Recent zbMATH articles in MSC 14https://www.zbmath.org/atom/cc/142021-03-30T15:24:00+00:00Werkzeug\(K3\) categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration.https://www.zbmath.org/1455.140802021-03-30T15:24:00+00:00"Shen, Junliang"https://www.zbmath.org/authors/?q=ai:shen.junliang"Yin, Qizheng"https://www.zbmath.org/authors/?q=ai:yin.qizhengSummary: We explore the connection between \(K3\) categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative \(K3\) category associated to a nonsingular cubic 4-fold.
By introducing a filtration on the \(\mathrm{CH}_1 \)-group of a cubic 4-fold \(Y\), we conjecture a sheaf/cycle correspondence for the associated \(K3\) category \(\mathcal{A}_Y \). This is a noncommutative analog of O'Grady's conjecture concerning derived categories of \(K3\) surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves.
Our method provides systematic constructions of (a) the Beauville-Voisin filtration on the \(\mathrm{CH}_0 \)-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in \(\mathcal{A}_Y \).On the derived category of \(\text{IGr}(3,8)\).https://www.zbmath.org/1455.140362021-03-30T15:24:00+00:00"Guseva, L. A."https://www.zbmath.org/authors/?q=ai:guseva.l-aA theorem of Tits type for automorphism groups of projective varieties in arbitrary characteristic. With an appendix by Tomohide Terasoma.https://www.zbmath.org/1455.140822021-03-30T15:24:00+00:00"Hu, Fei"https://www.zbmath.org/authors/?q=ai:hu.fei.1|hu.feiSummary: We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by \textit{J. Keum} et al. [Math. Res. Lett. 16, No. 1, 133--148 (2009; Zbl 1172.14025)] for complex projective varieties.Discriminant and Hodge classes on the space of Hitchin covers.https://www.zbmath.org/1455.140102021-03-30T15:24:00+00:00"Basok, Mikhail"https://www.zbmath.org/authors/?q=ai:basok.mikhailSummary: We continue the study of the rational Picard group of the moduli space of Hitchin spectral covers started in [\textit{D. Korotkin} and \textit{P. Zograf}, J. Math. Phys. 59, No. 9, 091412, 14 p. (2018; Zbl 1433.14023)]. In the first part of the paper we expand the ``boundary'', ``Maxwell stratum'' and ``caustic'' divisors introduced in [loc. cit.] via the set of standard generators of the rational Picard group. This generalizes the result of [loc. cit.], where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rational Picard group of the moduli space of Hitchin spectral covers.Deformations of Kolyvagin systems.https://www.zbmath.org/1455.111462021-03-30T15:24:00+00:00"Büyükboduk, Kâzım"https://www.zbmath.org/authors/?q=ai:buyukboduk.kazimSummary: Ochiai has previously proved that the Beilinson-Kato Euler systems for modular forms interpolate in nearly-ordinary \(p\)-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai's work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson-Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus), assuming that the deformation problem at hand is unobstructed in the sense of Mazur. We then use what we call \textit{universal Kolyvagin systems} to attempt a main conjecture over the eigencurve. Along the way, we utilize these objects in order to define a quasicoherent sheaf on the eigencurve that behaves like a \(p\)-adic \(L\)-function (in a certain sense of the word, in 3-variables).Refined \(\text{SU}(3)\) Vafa-Witten invariants and modularity.https://www.zbmath.org/1455.141052021-03-30T15:24:00+00:00"Göttsche, Lothar"https://www.zbmath.org/authors/?q=ai:gottsche.lothar"Kool, Martijn"https://www.zbmath.org/authors/?q=ai:kool.martijnSummary: We conjecture a formula for the refined \(\text{SU}(3)\) Vafa-Witten invariants of any smooth surface \(S\) satisfying \(H_1(S,\mathbb{Z})=0\) and \(p_g(S)>0\). The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined \(S\)-duality modularity transformation.
We provide evidence for our formula by calculating virtual \(\chi_y\)-genera of moduli spaces of rank \(3\) stable sheaves on \(S\) in examples using Mochizuki's formula. Further evidence is based on the recent definition of refined \(\text{SU}(r)\) Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank \(2)\) and Laarakker (rank \(3)\).Period relations for Riemann surfaces with many automorphisms.https://www.zbmath.org/1455.140632021-03-30T15:24:00+00:00"Candelori, Luca"https://www.zbmath.org/authors/?q=ai:candelori.luca"Fogliasso, Jack"https://www.zbmath.org/authors/?q=ai:fogliasso.jack"Marks, Christopher"https://www.zbmath.org/authors/?q=ai:marks.christopher"Moses, Skip"https://www.zbmath.org/authors/?q=ai:moses.skipSummary: By employing the theory of vector-valued automorphic forms for non-unitarizable representations, we provide a new bound for the number of linear relations with algebraic coefficients between the periods of an algebraic Riemann surface with many automorphisms. Previously, the strongest general bound for this number was the genus of the Riemann surface, a result due to Wolfart. Our new bound significantly improves on this estimate, and it can be computed explicitly from the canonical representation of the Riemann surface. This bound may then be used (as observed by Shiga and Wolfart) to estimate the dimension of the endomorphism algebra of the Jacobian of the Riemann surface. We demonstrate with a few examples how this improved bound allows one, in some instances, to actually compute the dimension of this endomorphism algebra and to thereby determine whether the Jacobian has complex multiplication.
For the entire collection see [Zbl 1452.17002].On classification of toric surface codes of dimension seven.https://www.zbmath.org/1455.140552021-03-30T15:24:00+00:00"Hussain, Naveed"https://www.zbmath.org/authors/?q=ai:hussain.naveed"Luo, Xue"https://www.zbmath.org/authors/?q=ai:luo.xue"Yau, Stephen S.-T."https://www.zbmath.org/authors/?q=ai:yau.stephen-shing-toung"Zhang, Mingyi"https://www.zbmath.org/authors/?q=ai:zhang.mingyi"Zuo, Huaiqing"https://www.zbmath.org/authors/?q=ai:zuo.huaiqingFollowing previous work on the classification of low-dimensional toric surface codes by \textit{S. S. T. Yau} and \textit{H. Zuo} [Appl. Algebra Eng. Commun. Comput. 20, No. 2, 175--185 (2009; Zbl 1174.94029)] and \textit{X. Luo} et al. [Finite Fields Appl. 33, 90--102 (2015; Zbl 1394.14017)] the main results of the paper under review extend the classification, up to monomial equivalence, of toric surface codes of dimension less than or equal to \(7\), with the monomial equivalence of some pairs of toric codes in this range still undetermined.
Reviewer: Felipe Zaldívar (Ciudad de México)Diagrams for nonabelian Hodge spaces on the affine line.https://www.zbmath.org/1455.140672021-03-30T15:24:00+00:00"Boalch, Philip"https://www.zbmath.org/authors/?q=ai:boalch.philip-p"Yamakawa, Daisuke"https://www.zbmath.org/authors/?q=ai:yamakawa.daisukeIn a previous paper of the first author [``Irregular connections and Kac-Moody root systems'', Preprint, \url{arXiv:0806.1050}], a diagram was defined for each algebraic connection (unramified irregular at infinity) on a vector bundle on the affine line. The diagram is a doubled quiver. The connection determines a single nonabelian Hodge space with a triple of distinct algebraic
structures thus allowing to attach a diagram to a class of nonabelian Hodge spaces. The present note extends this construction by defining a diagram for any algebraic connection on a vector bundle on the affine line, i.e. for any nonabelian Hodge space attached to the affine line. One can show using the Fourier-Laplace transform that any moduli space of meromorphic connections on a smooth affine curve of genus zero is isomorphic to one on the affine line (i.e. with just one puncture), and this is expected to hold for the full nonabelian Hodge triple.
Reviewer: Vladimir P. Kostov (Nice)Approximation of maps into spheres by piecewise-regular maps of class \(C^k\).https://www.zbmath.org/1455.141082021-03-30T15:24:00+00:00"Bilski, Marcin"https://www.zbmath.org/authors/?q=ai:bilski.marcinSummary: The aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise-regular maps of class \(\mathcal{C}^k,\) where \(k\) is an arbitrary nonnegative integer.Introductory course on \(\ell\)-adic sheaves and their ramification theory on curves.https://www.zbmath.org/1455.111562021-03-30T15:24:00+00:00"Kindler, Lars"https://www.zbmath.org/authors/?q=ai:kindler.lars"Rülling, Kay"https://www.zbmath.org/authors/?q=ai:rulling.kaySummary: These are the notes accompanying 13 lectures given by the authors at the Clay Mathematics Institute Summer School 2014 in Madrid. They give an introduction to the theory of \(\ell\)-adic sheaves with emphasis on their ramification theory on curves.
For the entire collection see [Zbl 1446.81001].Geometry of the Wiman-Edge pencil and the Wiman curve.https://www.zbmath.org/1455.140522021-03-30T15:24:00+00:00"Farb, Benson"https://www.zbmath.org/authors/?q=ai:farb.benson"Looijenga, Eduard"https://www.zbmath.org/authors/?q=ai:looijenga.eduard-j-nSummary: The \textit{Wiman-Edge pencil} is the universal family \(C_t,t\in\mathcal{B}\) of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group \(\mathfrak{A}_5\). The curve \(C_0\), discovered by
\textit{A. Wiman} [Stockh. Akad. Bihang \(\text{XXI}_1\). No. 3. 41 S. (1895; JFM 26.0658.03)] and called the \textit{Wiman curve}, is the unique smooth, genus 6 curve admitting a faithful action of the symmetric group \(\mathfrak{S}_5\). In this paper we give an explicit uniformization of \(\mathcal{B}\) as a non-congruence quotient \(\Gamma\backslash\mathfrak{H}\) of the hyperbolic plane \(\mathfrak{H}\), where \(\Gamma < \text{PSL}_2(\mathbb{Z})\) is a subgroup of index 18. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of \(C_t\) into 10 lines (resp. 5 conics) whose intersection graph is the Petersen graph (resp. \(K_5)\). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve \(C_0\) itself as the quotient \(\Lambda\backslash\mathfrak{H}\), where \(\Lambda\) is a principal level 5 subgroup of a certain ``unit spinor norm'' group of Möbius transformations. We then prove that \(C_0\) is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.Modified elliptic genus.https://www.zbmath.org/1455.810402021-03-30T15:24:00+00:00"Gritsenko, Valery"https://www.zbmath.org/authors/?q=ai:gritsenko.valerii-alekseevichSummary: This mini course is an additional part to my semester course on the theory of Jacobi modular forms given at the mathematical department of NRU HSE in Moscow (see the author Jacobi modular forms: 30 ans après; COURSERA (12 lectures and seminars), 2017--2019). This additional part contains some applications of Jacobi modular forms to the theory of elliptic genera and Witten genus. The subject of this course is related to my old talk given in Japan (see the author [RIMS Kokyuroku 1103, 71--85 (1999; Zbl 1015.11015)]). Our approach based on special representations of Jacobi theta-series founded in the author [loc. cit.] (see Lemma 1.5 below). In the next future we hope to present new automorphic invariants of holomorphic varieties at the automorphic seminar of the International laboratories of mirror symmetry and automorphic forms of NRU HSE. So this mini course will be also an introduction to this new theory. The elliptic genus of a Calabi-Yau manifold is an automorphic form in two variables with integral Fourier coefficients. More exactly this is a weak Jacobi form of weight zero. For arbitrary holomorphic vector bundle over a complex manifold we define an automorphic correction of its elliptic genus which is a meromorphic Jacobi form. We calculate this Jacobi form explicitly in the case of manifolds of small dimensions. Using arithmetic properties of the graded ring of integral Jacobi forms we obtain some geometrical applications.
For the entire collection see [Zbl 1454.81013].Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures).https://www.zbmath.org/1455.140222021-03-30T15:24:00+00:00"Mellit, Anton"https://www.zbmath.org/authors/?q=ai:mellit.anton-sThe author shows that Schiffmann's formulas for counts of Higgs bundles over finite fields can be reduced to a simpler formula conjedtured by Mozgovoy.
Consider the Frobenius action on the first cohomology of \(C\) with eigenvalues \(\alpha_1, \dots, \alpha_{2g}\) with \(\alpha_{i+g} = q\alpha_i^{-1}\) for \(i = 1, \dots, g\), which gives
\[
\# C(\mathbb{F}_{q^k}) = 1 + q^k - \sum_{i=1}^{2g} \alpha_i^k.
\]
Schiffmann's result in [\textit{R. Fedorov} et al., Commun. Number Theory Phys. 12, No. 4, 687--766 (2019; Zbl 1409.14059)] says that the number of absolutely indecomposable bundles or rank \(r\) and degree \(d\) over a complete curve \(C\) of genus \(g\) over \(\mathbb{F}_q\) is given by a Laurent polynomial
\[
A_{g,r,d} (q, \alpha_1, \dots, \alpha_{g} ) \in \mathbb{Z} [q, \alpha_1^{\pm 1}, \dots, \alpha_{g}^{\pm 1} ]
\]
which is independent of \(C\), symmetric in \(\alpha_i\), and invariant under \(\alpha_o \to q \alpha_i^{-1}\).
The main result of this paper is:
Theorem 1.1. Let \(g \ge 1\). Let \(\Omega_g\) denote the series
\[
\Omega_g = \sum_{\mu \in \mathcal{P}} T^{|\mu|} \prod_{\square \in \mu} \frac{ \prod_{i=1}^g (z^{a(\square)+1} - \alpha_i q^{l (\square)} ) (z^{a(\square)} - \alpha_i^{-1} q^{l (\square)+1} ) }{ (z^{a(\square)+1} - q^{l (\square)} ) (z^{a(\square)} - q^{l (\square)+1} ) },
\]
where \(a(\square)\), \(l(\square)\), and \(\mathcal{P}\) are certain combinatorial notations from Young diagrams. Also let
\[
H_g = -(1-q) (1-z) \operatorname{Log} \Omega_g, H_g = \sum_{r=1}^{\infty} H_{g,r} T^r.
\]
Then for all \(r \ge 1\), \(H_{g,r}\) is a Laurent polynomial in \(q,z\) and \(\alpha, \dots, \alpha_g\), and for all \(d\), \(A_{g,r,d}\) is obtained by setting \(z=1\) in \(H_{g,r}\):
\[
A_{g,r,d} (q, \alpha_1, \dots, \alpha_g) = H_{g,r} (q, 1, \alpha_1, \dots, \alpha_g).
\]
As a corollary, the GL-version of the conjecture of \textit{T. Hausel} (Conjecture 3.2 in [Prog. Math. 235, 193--217 (2005; Zbl 1099.14026)]) is obtained:
Corollary 1.2. For \(r, d, d'\) satisfying \((r,d) = (r,d') =1\), the \(E\)-polynomials of \(\mathcal{M}(g,r,d)\) and \(\mathcal{M}(g,r,d')\) coincide.
In the next paper [Ann. Math. (2) 192, No. 1, 165--228 (2020; Zbl 07239273)], the author extends the methods of [\textit{O. Schiffmann}, Ann. Math. (2) 183, No. 1, 297--362 (2016; Zbl 1342.14076)] to the parabolic case. This gives a proof of the conjecture of \textit{T. Hausel} et al. [Duke Math. J. 160, No. 2, 323--400 (2011; Zbl 1246.14063)] on the Poincaré polynomials of character varieties with punctures.
Reviewer: Insong Choe (Seoul)Betti numbers of MCM modules over the cone of an elliptic normal curve.https://www.zbmath.org/1455.140652021-03-30T15:24:00+00:00"Pavlov, Alexander"https://www.zbmath.org/authors/?q=ai:pavlov.aleksander|pavlov.alexander-a.1In [\textit{ A. Pavlov}, ``Betti tables of maximal Cohen-Macaulay modules over the cone of a plane cubic'', Preprint, \url{arXiv:1511.05089}] the author obtained formulas for the Betti numbers of maximal Cohen-Macaulay modules over the cone of a smooth plane cubic. In the present paper he extends his results to normal elliptic curves in $\mathbb{P}^n$, with $n>2$. He uses Orlov's equivalence of triangulated categories. See [\textit{D. Orlov}, Prog. Math. 270, 503--531 (2009; Zbl 1200.18007)]. and derives recurrence relations for the Betti numbers. Furthermore, he applies his methods to the special cases $n = 1, 2$ and derives formulas for the Betti numbers and the numerical invariants of maximal Cohen-Macaulay modules. For background see [\textit{J. Herzog} and \textit{M. Kühl}, Adv. Stud. Pure Math. 11, 65--92 (1987; Zbl 0641.13014)].
Reviewer: Aigli Papantonopoulou (Ewing)Isotopic meshing of a real algebraic space curve.https://www.zbmath.org/1455.141172021-03-30T15:24:00+00:00"Jin, Kai"https://www.zbmath.org/authors/?q=ai:jin.kai"Cheng, Jinsan"https://www.zbmath.org/authors/?q=ai:cheng.jinsanSummary: This paper presents a new algorithm for computing the topology of an algebraic space curve. Based on an efficient weak generic position-checking method and a method for solving bivariate polynomial systems, the authors give a first deterministic and efficient algorithm to compute the topology of an algebraic space curve. Compared to extant methods, the new algorithm is efficient for two reasons. The bit size of the coefficients appearing in the sheared polynomials are greatly improved. The other is that one projection is enough for most general cases in the new algorithm. After the topology of an algebraic space curve is given, the authors also provide an isotopic-meshing (approximation) of the space curve. Moreover, an approximation of the algebraic space curve can be generated automatically if the approximations of two projected plane curves are first computed. This is also an advantage of our method. Many non-trivial experiments show the efficiency of the algorithm.Visualizing planar and space implicit real algebraic curves with singularities.https://www.zbmath.org/1455.650832021-03-30T15:24:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuan"Feng, Yong"https://www.zbmath.org/authors/?q=ai:feng.yongSummary: This paper presents a new method for visualizing implicit real algebraic curves inside a bounding box in the 2-D or 3-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of a curve near a singular point of it is not numerically stable, the authors trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is e-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the projection of the curve in \(\mathbb{R}^2\), which is important for applications such as solving bi-parametric polynomial systems. The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small spheres centered at singular points, regular critical points of every connected components of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters, tracing the curve by a try-and-resume strategy and handling ``pseudo singular points''. The effectiveness of the algorithm is illustrated by numerous examples. This manuscript extends the proposed preliminary results that appeared in CASC 2018.The motivic nearby fiber and degeneration of stable rationality.https://www.zbmath.org/1455.140292021-03-30T15:24:00+00:00"Nicaise, Johannes"https://www.zbmath.org/authors/?q=ai:nicaise.johannes"Shinder, Evgeny"https://www.zbmath.org/authors/?q=ai:shinder.evgenyIn the article under review are investigated the specialization properties of stable rationality in regular families over a field of characteristic 0. For such a field \(k\), define \(R = k[[t]], K = k((t)), K(\infty) = \bigcup_n k((t^{\frac{1}{n}}))\). Let \(\hat{\mu}\) be the projective limit of the groups of roots of unity.
For a proper scheme \(X/K\), an \(R\)-model of \(X\) is flat and proper scheme \(\mathcal{X}/R\) with an isomorphism \(\mathcal{X}_K \simeq X\). For smooth \(X\), \(\mathcal{X}\) is called strict normal crossings (snc) if it is regular, and the special fiber is snc divisor, so it can be represented as \(\mathcal{X}_k = \sum_i N_{i}E_i\).
\(\mathbf{K}(\mathrm{Var}_K)\) is the Grothendieck ring of varieties over \(K\), and \(\mathbf{K}^{\hat{\mu}}(\mathrm{Var}_K)\) is the Grothendieck ring of varieties over \(K\) with \(\hat{\mu}\)-action. It is generated by the classes of \(\hat{\mu}\)-equivariant isomorphisms of schemes of finite type with a good \(\hat{\mu}\)-action.
The first main result relies on the theorems of \textit{F. Bittner} [Compos. Math. 140, No. 4, 1011--1032 (2004; Zbl 1086.14016)] and \textit{M. Larsen} and \textit{V. A. Lunts} [Mosc. Math. J. 3, No. 1, 85--95 (2003; Zbl 1056.14015)], and it claims that stable rationality specializes in regular families whose fibers have at worst rational double points as singularities. Aiming to control stable rationality in families, two specialization morphisms are defined: the motivic volume \(\mathrm{Vol}_K: \mathbf{K}^G(\mathrm{Var}_K) \rightarrow \mathbf{K}^{\hat{\mu}}(\mathrm{Var}_k) \), inspired by [\textit{J. Denef} and \textit{F. Loeser}, Prog. Math. 201, 327--348 (2001; Zbl 1079.14003); \textit{E. Hrushovski} and \textit{D. Kazhdan}, Prog. Math. 253, 261--405 (2006; Zbl 1136.03025)], and the motivic reduction \(MR: \mathbf{K}(\mathrm{Var}_K) \rightarrow \mathbf{K}(Var_k)\). For both are obtained formulas for given snc model of proper and smooth scheme. As application is shown that for proper and smooth \(K(\infty)\)-schemes \(X, Y\), which are stably birational, \(\mathrm{Vol}(X) \equiv\mathrm{Vol}(Y)\bmod \mathbb{L}\). From it follows that having \(\mathcal{X}, \mathcal{Y}\) proper smooth \(R\)-schemes with \(\mathcal{X}_{K(\infty)}\), \(\mathcal{Y}_{K(\infty)} \) stably birational, then \(\mathcal{X}_k \) is stably birational to \(\mathcal{Y}_k \). This is generalized then to strict normal crossings degenerations. As a corollary, over uncountable algebraically closed \(k\) and \(S\) of finite type, given proper and smooth morphism \(f: X \rightarrow S\) whose very general closed fiber is stably rational then every closed fiber of \(f\) is stably rational. This is generalized further to regular \(R\)-models \(\mathcal{X}\) whose special fiber is reduced with at worst ordinary double points as singularities. It is proved that \(\mathcal{X}\) is then \(\mathbb{L}\)-faithful, that is, \([\mathcal{X}_k] \equiv \mathrm{Vol}(X_{K(\infty)})\pmod \mathbb{L}\).
The second main result in the article is about proper flat families \(f:\mathcal{X} \rightarrow C\) of connected smooth schemes over a curve \(C\), with the geometric generic fiber of \(f\) stably rational. It asserts that any geometric fiber with at most ordinary double points has stably rational component. As an application is shown that a very general smooth quartic threefolds are not stably rational (see [\textit{J.-L. Colliot-Thélène} and \textit{A. Pirutka}, Ann. Sci. Éc. Norm. Supér. (4) 49, No. 2, 371--397 (2016; Zbl 1371.14028)]).
In the appendix are proved the existence of the motivic volume with a formula in terms of smooth logarithmic models, generalizing the formula in the main text, and a refined version of the existence of the motivic reduction.
The arguments used in the article under review are strengthened further in [\textit{M. Kontsevich} and \textit{Yu. Tschinkel}, ``Specialization of birational types'', Preprint, \url{arXiv:1708.05699}]. There are constructed specialization homomorphisms analogous to motivic volume and motivic reduction, with the Grothendieck ring replaced by the Burnside ring of \(k\), which is a new invariant. As a result is obtained an important generalization of the first main result above from stable rationality to rationality. Nevertheless the method in the present paper remains of independent interest.
Reviewer: Peter Petrov (Sofia)Bounded automorphism groups of compact complex surfaces.https://www.zbmath.org/1455.140832021-03-30T15:24:00+00:00"Prokhorov, Yu. G."https://www.zbmath.org/authors/?q=ai:prokhorov.yuri-g"Shramov, C. A."https://www.zbmath.org/authors/?q=ai:shramov.konstantin-aSingularities of normal log canonical del Pezzo surfaces of rank one.https://www.zbmath.org/1455.140742021-03-30T15:24:00+00:00"Kojima, Hideo"https://www.zbmath.org/authors/?q=ai:kojima.hideoSummary: Let \(X\) be a normal del Pezzo surface of rank one with only rational log canonical singular points. In this paper, we prove that \(X\) can have at most one non klt singular point.
For the entire collection see [Zbl 07190056].Residual coordinates over one-dimensional rings.https://www.zbmath.org/1455.130152021-03-30T15:24:00+00:00"El Kahoui, M'hammed"https://www.zbmath.org/authors/?q=ai:el-kahoui.mhammed"Essamaoui, Najoua"https://www.zbmath.org/authors/?q=ai:essamaoui.najoua"Ouali, Mustapha"https://www.zbmath.org/authors/?q=ai:ouali.mustaphaSummary: Given a noetherian ring \(R\) and \(n\geq 2\), it is well-known that residual coordinates of the polynomial algebra \(R^{[n]}\) are \(m\)-stable coordinates for some \(m\geq 1\), that is they become coordinates in the larger polynomial algebra \(R^{[n+m]}\). In this paper we prove that, over a large class of noetherian one-dimensional rings, \(m=1\) is enough. This includes affine algebras over an algebraically closed field as well as noetherian complete local rings containing a field.What is \(\ldots\) a hyperpolygon?https://www.zbmath.org/1455.140232021-03-30T15:24:00+00:00"Rayan, Steven"https://www.zbmath.org/authors/?q=ai:rayan.steven"Schaposnik, Laura P."https://www.zbmath.org/authors/?q=ai:schaposnik.laura-p(no abstract)Toric log del Pezzo surfaces with one singularity.https://www.zbmath.org/1455.140982021-03-30T15:24:00+00:00"Dais, Dimitrios I."https://www.zbmath.org/authors/?q=ai:dais.dimitrios-iSummary: This paper focuses on the classification (up to isomorphism) of all toric log Del Pezzo surfaces with exactly one singularity, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective spaces.Gaps in the number of generators of monomial Togliatti systems.https://www.zbmath.org/1455.130332021-03-30T15:24:00+00:00"Almeida, Charles"https://www.zbmath.org/authors/?q=ai:almeida.charles"Andrade, Aline V."https://www.zbmath.org/authors/?q=ai:andrade.aline-v"Miró-Roig, Rosa M."https://www.zbmath.org/authors/?q=ai:miro-roig.rosa-mariaA Togliatti system is a homogeneous ideal \(I\) of the polynomial ring \(R=k[x_0,\ldots,x_n]\), minimally generated by \(\mu(I)\) polynomials all of the same degree \(d\), such that the quotient ring \(R/I\) is artinian and \(I\) fails the weak Lefschetz property in degree \(d-1\), i.e., there exists a linear form \(L\) such that the map of multiplication by \(L\) from \((R/I)_{d-1}\) to \((R/I)_d\) is not of maximal rank. Togliatti systems have been introduced in the article [\textit{E. Mezzetti} et al., Can. J. Math. 65, No. 3, 634--654 (2013; Zbl 1271.13036)], where it has been shown that they are connected via apolarity to projective varieties satisfying Laplace equations.
A Togliatti system is minimal if there is no proper subset of its set of generators defining again a monomial Togliatti system.
Aim of the article under review is to find which numbers can appear as \(\mu(I_{d,n})\) for a minimal monomial Togliatti system \(I_{d,n}\), with given \(n\) and \(d\). It is known that \(2n+1\leq\mu(I_{d,n})\leq \binom{d+n-1}{n-1}\) (see [\textit{E. Mezzetti} and \textit{R. M. Miró-Roig}, Ann. Mat. Pura Appl. (4) 195, No. 6, 2077--2098 (2016; Zbl 1357.13024)]). Here the authors prove that there are gaps in the above interval, precisely for any \(n \geq 4\) and \(d \geq 3\), the integer values between \(2n + 3\) and \(3n-1\) cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. On the other hand, in the case \(n=4\), for any \(d\geq 3\), they prove that all the integer numbers in the interval \([9,\binom{d+3}{3}]\) except \(11\), can be realized.
The proofs consist in an accurate analysis of the configuration of the finite subsets in \(\mathbb Z_{n+1}\) corresponding to monomials of degree \(d\) in the Macaulay inverse system, relying on the combinatorial characterization of monomial Togliatti systems. The article is completed and illustrated by many examples and counterexamples.
Reviewer: Emilia Mezzetti (Trieste)Maximally-graded matrix factorizations for an invertible polynomial of chain type.https://www.zbmath.org/1455.140332021-03-30T15:24:00+00:00"Aramaki, Daisuke"https://www.zbmath.org/authors/?q=ai:aramaki.daisuke"Takahashi, Atsushi"https://www.zbmath.org/authors/?q=ai:takahashi.atsushi.3|takahashi.atsushi.2Summary: \textit{P. Orlik} and \textit{R. Randell} [Invent. Math. 39, 199--211 (1977; Zbl 0341.14001)] constructed a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles. The purpose of this paper is to prove the algebraic counterpart of the Orlik-Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the [\textit{P. Berglund} and \textit{T. Hübsch}, Nucl. Phys., B 393, No. 1--2, 377--391 (1993; Zbl 1245.14039)] transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of \textit{A. Kuznetsov} and \textit{M. Smirnov} [Proc. Lond. Math. Soc. (3) 120, No. 5, 617--641 (2020; Zbl 07194979)], whose Euler matrix are calculated in terms of the ``zeta function'' of the inverse of the polarization.
As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).On the problem of resolution of singularities: its development and current status (from a myopic point of view of one researcher struggling with the problem).https://www.zbmath.org/1455.140032021-03-30T15:24:00+00:00"Matsuki, Kenji"https://www.zbmath.org/authors/?q=ai:matsuki.kenjiFrom the introduction: The goal of this article is to give a mathematical formulation and explanation of the problem of resolution of singularities, and at the same time, although we start with the basic introduction for the beginner, we try to report on the recent developments and current status of the problem at the research front.2-nilpotent co-Higgs structures.https://www.zbmath.org/1455.140842021-03-30T15:24:00+00:00"Ballico, Edoardo"https://www.zbmath.org/authors/?q=ai:ballico.edoardo"Huh, Sukmoon"https://www.zbmath.org/authors/?q=ai:huh.sukmoonA pair \((E,\phi)\) formed by a vector bundle \(E\) on a smooth variety \(X\) and a morphism \(\phi\colon E \rightarrow E \otimes T_X\) satisfying \(\phi \wedge \phi=0\) is called a co-Higgs bundle and \(\phi\) is called a co-Higgs field.
Making clever use of the Hartshorne-Serre correspondence, the authors obtain existence of nilpotent co-Higgs structures on both strictly semistable indecomposable bundles and stable bundles on \(\mathbb P^3\) for any positive value of \(c_2\).
Furthermore, they show that the Chern numbers of any nonsplit strictly semistable rank 2 bundle on \(\mathbb P^3\) with a nilpotent co-Higgs structure satisfy \(4c_2-c_1>32\).
Reviewer: Elizabeth Gasparim (Antofagasta)Algorithms for solving an algebraic equation.https://www.zbmath.org/1455.682812021-03-30T15:24:00+00:00"Bruno, A. D."https://www.zbmath.org/authors/?q=ai:bruno.alexander-dSummary: For finding global approximate solutions to an algebraic equation in \(n\) unknowns, the Hadamard open polygon for the case \(n = 1\) and Hadamard polyhedron for the case \(n = 2\) are used. The solutions thus found are transformed to the coordinate space by a translation (for \(n = 1)\) and by a change of coordinates that uses the curve uniformization (for \(n = 2)\). Next, algorithms for the local solution of the algebraic equation in the vicinity of its singular (critical) point for obtaining asymptotic expansions of one-dimensional and two-dimensional branches are presented for \(n = 2\) and \(n = 3\). Using the Newton polygon (for \(n = 2)\), the Newton polyhedron (for \(n = 3)\), and power transformations, this problem is reduced to situations similar to those occurring in the implicit function theorem. In particular, the local analysis of solutions to the equation in three unknowns leads to the uniformization problem of a plane curve and its transformation to the coordinate axis. Then, an asymptotic expansion of a part of the surface under examination can be obtained in the vicinity of this axis. Examples of such calculations are presented.Orbits of real semisimple Lie groups on real loci of complex symmetric spaces.https://www.zbmath.org/1455.140992021-03-30T15:24:00+00:00"Cupit-Foutou, Stéphanie"https://www.zbmath.org/authors/?q=ai:cupit-foutou.stephanie"Timashev, Dmitry A."https://www.zbmath.org/authors/?q=ai:timashev.dmitri-aSummary: Let \(G\) be a complex semisimple algebraic group and \(X\) be a complex symmetric homogeneous \(G\)-variety. Assume that both \(G\), \(X\) as well as the \(G\)-action on \(X\) are defined over real numbers. Then \(G(\mathbb{R})\) acts on \(X(\mathbb{R})\) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of \textit{A. Borel} and \textit{L. Ji} [Compactifications of symmetric and locally symmetric spaces. Basel: Birkhäuser (2006; Zbl 1100.22001)].Singularities and genus of the \(k\)-ellipse.https://www.zbmath.org/1455.141102021-03-30T15:24:00+00:00"Jiang, Yuhan"https://www.zbmath.org/authors/?q=ai:jiang.yuhan"Han, Weiqiao"https://www.zbmath.org/authors/?q=ai:han.weiqiaoSummary: A \(k\)-ellipse is a plane curve consisting of all points whose distances from \(k\) fixed foci sum to a constant. We determine the singularities and genus of its Zariski closure in the complex projective plane. The paper resolves an open problem stated by \textit{J. Nie} et al. [IMA Vol. Math. Appl. 146, 117--132 (2008; Zbl 1171.90498)].Erratum to: Mukai's program (reconstructing a \(K3\) surface from a curve) via wall-crossing.https://www.zbmath.org/1455.140352021-03-30T15:24:00+00:00"Feyzbakhsh, Soheyla"https://www.zbmath.org/authors/?q=ai:feyzbakhsh.soheylaFrom the text: There is a mistake in the proof of [the author, J. Reine Angew. Math. 765, 101--137 (2020; Zbl 1455.14034), Proposition 5.2 (a), line 5]. Therefore, the proof is valid only for a certain range of curves. The author treats the missing cases in [Mukai's program (reconstructing a \(K3\) surface from a curve) via wall-crossing, II, preprint, \url{arXiv:2006.08410}].A motivic Segal theorem for pairs (announcement).https://www.zbmath.org/1455.140442021-03-30T15:24:00+00:00"Tsybyshev, A."https://www.zbmath.org/authors/?q=ai:tsybyshev.a-eSummary: In order to provide a new, more computation-friendly, construction of the stable motivic category \(SH (k)\), V. Voevodsyky laid the foundation of delooping motivic spaces.
\textit{G. Garkusha} and \textit{I. Panin} [``Framed motives of algebraic varieties (after V. Voevodsky)'', Preprint, \url{arXiv:1409.4372}] based on joint works with
\textit{G. Garkusha} et al. [``Framed motives of relative motivic spheres'', Preprint, \url{arXiv:1604.02732}], \textit{A. Ananyevskiy} et al. [``Cancellation theorem for framed motives of algebraic varieties'', Preprint, \url{arXiv:1601.06642}], \textit{A. Druzhinin} and \textit{I. Panin} [``Surjectivity of the etale excision map for homotopy invariant framed presheaves'', Preprint, \url{arXiv:1808.07765}] made that project a reality. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field \(k\) and any \(k\)-smooth scheme \(X\), the canonical morphism of motivic spaces \({C}_{\ast } Fr(X)\to{\Omega}_{{\mathbb{P}}^1}^{\infty }{\sum}_{{\mathbb{P}}^1}^{\infty}\left({X}_+\right)\) is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs \((X,U)\), where \(X\) is a \(k\)-smooth scheme and \(U\) is its open subscheme intersecting each component of \(X\) in a nonempty subscheme. It is claimed that in this case the motivic space \(C_* Fr ((X,U))\) is a Nisnevich locally connected, and the motivic space morphism \({C}_{\ast } Fr\left(\left(X,U\right)\right)\to{\Omega}_{{\mathbb{P}}^1}^{\infty }{\sum}_{{\mathbb{P}}^1}^{\infty}\left(X/U\right)\) is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of \(S = X -U\) in each component of \(X\) is greater than \(r \geq 0\), then the simplicial sheaf \(C_* Fr ((X,U))\) is locally \(r\)-connected.Self-dual intervals in the Bruhat order.https://www.zbmath.org/1455.050792021-03-30T15:24:00+00:00"Gaetz, Christian"https://www.zbmath.org/authors/?q=ai:gaetz.christian"Gao, Yibo"https://www.zbmath.org/authors/?q=ai:gao.yiboSummary: \textit{A. Björner} and \textit{T. Ekedahl} [Ann. Math. (2) 170, No. 2, 799--817 (2009; Zbl 1226.05268)] prove that general intervals \([e, w]\) in Bruhat order are ``top-heavy'', with at least as many elements in the \(i\)-th corank as the \(i\)-th rank. Well-known results of \textit{J. B. Carrell} [Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)] and of \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45--52 (1990; Zbl 0714.14033)] give the equality case: \([e, w]\) is rank-symmetric if and only if the permutation \(w\) avoids the patterns 3412 and 4231 and these are exactly those \(w\) such that the Schubert variety \(X_w\) is smooth. In this paper we study the finer structure of rank-symmetric intervals \([e, w]\), beyond their rank functions. In particular, we show that these intervals are still ``top-heavy'' if one counts cover relations between different ranks. The equality case in this setting occurs when \([e, w]\) is self-dual as a poset; we characterize these \(w\) by pattern avoidance and in several other ways.Strong \((\delta,n)\)-complements for semi-stable morphisms.https://www.zbmath.org/1455.140302021-03-30T15:24:00+00:00"Filipazzi, Stefano"https://www.zbmath.org/authors/?q=ai:filipazzi.stefano"Moraga, Joaquín"https://www.zbmath.org/authors/?q=ai:moraga.joaquinLet \(X\) be a normal quasi-projective variety over an algebraically closed field \(k\) of characteristic \(0\), and \(B\) an effective Weil divisor on \(X\). Let \(\pi\colon X' \rightarrow X\) be a projective birational morphism and let \(M = \pi_*M'\) for \(M'\) a \(\mathbb{Q}\)-Cartier nef divisor on \(X'\), such that \(K_X + B + M\) is \(\mathbb{Q}\)-Cartier. We say \((X, B+M)\) is a generalized pair, with boundary part \(B\) and moduli part \(M\). Suppose the generalized pair \((X,B+M)\) has \(\epsilon\)-log canonical singularities, and let \(X \rightarrow Z\) be a contraction of normal quasi-projective varieties. A Weil divisor \(B^+\) on \(X\) is a strong \((\delta,n)\)-complement over \(z \in Z\) if the following conditions hold over some neighborhood of \(z\): (a) \((X,B^+ + M)\) is a \(\delta\)-log canonical generalized pair with boundary part \(B^+\); (b) \(n(K_X + B^+ + M) \sim 0\); and (c) \(nB^+ \geq nB\).
\textit{C. Birkar} [Ann. Math. (2) 190, No. 2, 345--463 (2019; Zbl 07107180)] proved boundedness of strong \((0,n)\)-complements for log canonical pairs, which he used in the proof of the BAB conjecture. In the paper under review, the authors conjecture a form of boundedness of strong \((\delta,n)\)-complements for generalized pairs:
Conjecture 1: Let \(d,p \in \mathbb{N}\), \(\epsilon \in [0,1)\), and \(\Lambda \subset \mathbb{Q}\) a set satisfying the ACC condition with rational accumulation points. There exist \(n \in \mathbb{N}\) and \(\delta \in \mathbb{R}_{>0}\) only depending on \(d\), \(p\), \(\epsilon\) and \(\Lambda\) satisfying the following. Let \(X \rightarrow Z\) be a contraction between normal quasi-projective varieties, and \((X,B+M)\) be a generalized \(\epsilon\)-log canonical pair of dimension \(d\) such that: (1) \(-(K_X + B + M)\) is nef over \(Z\); (2) \(X\) is of Fano type over \(Z\); (3) \(\mathrm{coeff}(B) \subset \Lambda\); and (4) \(pM'\) is Cartier (notation as above).
Then, for every point \(z \in Z\), there exists a strong \((\delta,n)\)-complement for \((X,B+M)\) over \(z\). Moreover we can pick \(\delta > 0\) if \(\epsilon > 0\).
In this paper, the authors prove conjecture 1 if \(\delta = \epsilon = 0\), and in case \(Z = Spec(k)\), they prove it if \(\epsilon = 0\), or \(M'\) is trivial, or \(\Lambda\) is finite, and in any case they can take \(\delta = \epsilon\). They also prove the conjecture, fixing \(m \in \mathbb{N}\), if \(\epsilon > 0\), \(\Lambda\) is finite, \(M'\) is trivial, \(mK_Z\) is Cartier and \(X \rightarrow Z\) is a semi-stable morphism for the pair \((X,B)\). In this case, \(\delta > 0\) depends on \(m\) as well.
They also give some applications of these theorems. The first one is an effective version of the generalized canonical bundle formula, and also an effective version of generalized adjunction to exceptional generalized log canonical centers. Finally, they show that conjecture 1 implies a conjecture due to McKernan concerning the singularities of the base a Mori Fiber Space.
Reviewer: João Paulo Figueredo (São Paulo)The moduli space of twisted canonical divisors.https://www.zbmath.org/1455.140562021-03-30T15:24:00+00:00"Farkas, Gavril"https://www.zbmath.org/authors/?q=ai:farkas.gavril"Pandharipande, Rahul"https://www.zbmath.org/authors/?q=ai:pandharipande.rahulÉtant donné \(g\geq1\) et une partition \(\mu=(m_{1},\dots,m_{n})\) de \(2g-2\), on définit la strate \(\mathcal{H}(\mu)\) paramétrant les paires \((X;\omega)\) où \(X\) est une surface de Riemann de genre \(g\) et \(\omega\) est une différentielle abélienne telle que \(\operatorname{div}(\omega)=\sum m_{i}p_{i}\). Dans la suite, les différentielles sont considérées modulo l'action de \(\mathbb{C}^{\ast}\) par multiplication. Le but de cet article est de construire un espace compacte, l'espace des modules des diviseurs canoniques twistés, tel que la strate est un ouvert dense de l'une des composantes irréductibles.
Un diviseur canoniques twisté est une courbe pointé stable telle que la restriction à chaque composante des points avec multiplicité \(\mu\) est un diviseur canonique modulo une somme pondérée (par les twistes) des points nodaux de la composante. L'espace des modules des diviseurs canoniques twistés contient la strate comme un sous espace dense, et d'autres composantes contenues dans le bord de \(\overline{\mathcal{M}}_{g,n}\). Les auteurs étudient la dimension de ces composantes ainsi que la relation entre les diviseurs canoniques twistés et les structures spin introduites dans [\textit{M. Cornalba}, in: Proceedings of the first college on Riemann surfaces held in Trieste (1989; Zbl 0800.14011)]. Il est à noter que les points de la composante contenant la strate ont étés décrit dans [\textit{M. Bainbridge} et al., Duke Math. J. 167, No. 12, 2347--2416 (2018; Zbl 1403.14058)].
Enfin l'article contient un appendice de Félix Janda, Rahul Pandharipande, Aaron Pixton, and Dimitry Zvonkine. Ils énoncent, dans le cas des strates de différentielles abéliennes méromorphes, une conjecture explicite entre les classes fondamentales de toutes les composantes de l'espace des modules et une formule de Pixton. Cela leur permet de donner une formule dans le groupe de Chow pour la classe de la fermeture des strates dans l'espace des modules des diviseurs canoniques twistés (dans le cas méromorphe et holomorphe). Cette conjecture a été prouvée dans [{\textit Y. Bae} et al., ``Pixton's formula and Abel-Jacobi theory on the Picard stack'', Preprint, \url{arXiv:2004.08676}] où elle apparaît comme le théorème 9.
Reviewer: Quentin Gendron (Guanajuato)Finite group schemes and \(p\)-divisible groups.https://www.zbmath.org/1455.140892021-03-30T15:24:00+00:00"Oort, Frans"https://www.zbmath.org/authors/?q=ai:oort.frans(no abstract)Torsion group schemes as iterative differential Galois groups.https://www.zbmath.org/1455.120062021-03-30T15:24:00+00:00"Maurischat, Andreas"https://www.zbmath.org/authors/?q=ai:maurischat.andreasThe paper is devoted to the development of Galois theory for extensions of fields of nonzero characteristic. For such fields, one can construct (see [\textit{A. Maurischat}, Trans. Am. Math. Soc. 362, No. 10, 5411--5453 (2010; Zbl 1250.13009)]) a Galois theory similar to Kolchin's theory for extensions of differential fields, if one replaces ordinary derivations by the so-called higher derivations. Since the fields of constants arising in this case are usually not algebraically closed, finite group schemes are used instead of algebraic groups. The paper contains the main concepts and basic results of this theory as an introduction. Its ``main part is to find computable criteria when higher derivations are iterative derivations, and furthermore when an iterative derivation on the function field of an abelian variety is compatible with the addition map''. For the inverse problem of Galois theory, it is shown ``that torsion group schemes of abelian varieties in positive characteristic occur as iterative differential Galois groups of extensions of iterative differential fields''.
Reviewer: Mykola Grygorenko (Kyïv)Gamma functions, monodromy and Frobenius constants.https://www.zbmath.org/1455.140192021-03-30T15:24:00+00:00"Bloch, Spencer"https://www.zbmath.org/authors/?q=ai:bloch.spencer-j"Vlasenko, Masha"https://www.zbmath.org/authors/?q=ai:vlasenko.masha\textit{V. V. Golyshev} and \textit{D. Zagier} [Izv. Math. 80, No. 1, 24--49 (2016; Zbl 1369.14054); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 1, 27--54 (2016)] have introduced Frobenius constants \(\kappa_{\rho,n}\) associated to an ordinary linear differential operator \(L\) with a reflection type singularity at \(t=c\). For every other regular singularity \(t=c'\) and a homotopy class of paths \(\gamma\) joining \(c'\) and \(c\), constants \(\kappa_{\rho,n}(\gamma)\) describe the variation around \(c\) of the Frobenius solutions to \(L\) defined near \(t=c'\) and continued analytically along \(\gamma\). The purpose of this work is to develop a theory (suggested by Golyshev) of motivic Mellin transforms or motivic gamma functions. The main result relates the generating series \(\sum_{n=0}^{\infty}\kappa_{\rho,n}(s-\rho)^n\) to the Taylor expansion at \(s=\rho\) of a generalized gamma function, which is a Mellin transform of a solution of the differential operator dual to \(L\). It follows from this that the numbers \(\kappa_{\rho,n}\) are always periods when \(L\) is a geometric differential operator.
Reviewer: Vladimir P. Kostov (Nice)Finiteness properties of affine Deligne-Lusztig varieties.https://www.zbmath.org/1455.140532021-03-30T15:24:00+00:00"Hamacher, Paul"https://www.zbmath.org/authors/?q=ai:hamacher.paul"Viehmann, Eva"https://www.zbmath.org/authors/?q=ai:viehmann.evaSummary: Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of \(G\)-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.Descent for coherent sheaves along formal/open coverings.https://www.zbmath.org/1455.140122021-03-30T15:24:00+00:00"Hörmann, Fritz"https://www.zbmath.org/authors/?q=ai:hormann.fritzSummary: For a regular Noetherian scheme \(X\) with a divisor with strict normal crossings \(D\) we prove that coherent sheaves satisfy descent w.r.t. the ``covering'' consisting of the open parts in the various completions of \(X\) along the components of \(D\) and their intersections.Physics and geometry of knots-quivers correspondence.https://www.zbmath.org/1455.570162021-03-30T15:24:00+00:00"Ekholm, Tobias"https://www.zbmath.org/authors/?q=ai:ekholm.tobias"Kucharski, Piotr"https://www.zbmath.org/authors/?q=ai:kucharski.piotr"Longhi, Pietro"https://www.zbmath.org/authors/?q=ai:longhi.pietroThe recently proposed knots-quivers correspondence relates colored HOMFLY polynomials of a knot \(K\subset S^3\) to the representation theory of the knot quiver \(Q_K\). The evidence so far is computational: knot and quiver data are independently computed and then matched. In this paper the authors give a conjectural layout of matching physical and geometric objects on the two sides of the correspondence.
Via the large \(N\) duality colored HOMFLY polynomials are related to open Gromov-Witten invariants of the holomorphic curves bordering a deformation \(L_K\) of the knot conormal bundle in the resolved conifold \(X\). Via \(M\)-theory those invariants can in turn be related to a 3D \(\mathcal{N}=2\) gauge theory \(T[L_K]\) on \(S^1\times\mathbb{R}^2\). The authors introduce a novel dual description of \(T[L_K]\), which is an Abelian Chern-Simons matter theory \(T[Q_K]\) with \(U(1)\) gauge group and a single fundamental chiral field for each node of \(Q_K\). On the other side, \(T[Q_K]\) has a quiver description in terms of BPS vortices, and its BPS vortex spectrum is an \(S^1\times\mathbb{R}^2\) shadow (effective description) of a theory of M2 branes ending on M5 branes. \(Q_K\) itself is its shadow on \(X\), and describes M2 branes stretched near \(L_K\).
In geometric terms, the authors conjecture that the spectrum of holomorphic curves in \(X\) bordering \(L_K\) is generated by a finite number of framed basic disks. For simple knots the basic disks correspond to the monomials in the colored HOMFLY polynomial, but in general the picture is more complicated. The authors suggest that the disks can be obtained by degenerating \(L_K\) when the braid representative of \(K\) is degenerated into the unknot. The disks correspond to the nodes of the quiver \(Q_K\), and each sector of \(T[Q_K]\) corresponds to an ordinary \(U(1)\) Chern-Simons theory deformed by holomorphic curves that contain at least one copy of a fixed basic disk. The Chern-Simons couplings encoded in the quiver arrows correspond to linking and self-linking numbers of the disk boundaries as curves in \(L_K\).
As a result, the quiver partition function recovers the counts of holomorphic curves constructed from linked configurations of the basic disks. Analytically, an explicit change of variables converts it into the refined Gromov-Witten partition function (the one where the coefficient of \(x^n\) is the Poincaré polynomial of the HOMFLY homology) of holomorphic curves in \(X\) bordering \(L_K\). A precise geometric definition of this refined partition function is the subject of current research. The conjecture is illustrated with the examples of the unknot and the trefoil.
Reviewer: Sergiy Koshkin (Houston)On Fano schemes of linear spaces of general complete intersections.https://www.zbmath.org/1455.140932021-03-30T15:24:00+00:00"Bastianelli, Francesco"https://www.zbmath.org/authors/?q=ai:bastianelli.francesco"Ciliberto, Ciro"https://www.zbmath.org/authors/?q=ai:ciliberto.ciro"Flamini, Flaminio"https://www.zbmath.org/authors/?q=ai:flamini.flaminio"Supino, Paola"https://www.zbmath.org/authors/?q=ai:supino.paolaLet \(F_k(X)\subset \mathbb G(k, n)\) the \textit{Fano scheme} parameterizing \(k\)-dimensional linear subspaces contained in a variety \(X \subset \mathbb P^n\).
The paper under review concerns the case in which \(X\) is a complete intersection of multi-degree \(\mathbf d := (d_1 ,\dotsc , d_s )\), with
\(1\le s \le n - 2\) and \(\prod_{i=1}^s d_i\ge 2\) and the main result of it extends a recent result by Riedl and Yang on the case of hypersurfaces (see [\textit{E. Riedl} and \textit{D. Yang}, J. Differ. Geom. 116, No. 2, 393--403 (2020; Zbl 1448.14042)], Theorem 3.3) which asserts that a very general hypersurface \(X\subset \mathbb P^n\) of degree \(d\) such that \(n \le \frac{(d+1)(d+2)}{6}\), then \(F_1 (X)\) contains no rational curves.
The result is the following (Theorem 1.2): Let \(X \subset \mathbb P^n\) be a very general complete intersection of multi-degree \(\mathbf d := (d_1 ,\dotsc , d_s )\), with \(1\le s \le n - 2\) and \(\prod_{i=1}^s d_i> 2\). Let \(1\le k\le n - s - 1\) be an integer. If
\[n\le k-1+\frac{1}{k+2}\sum_{i=1}^s\binom{d_i+k+1}{k-1},\]
then \(F_k (X)\) contains neither rational nor elliptic curves.
Moreover, in Section 4, Theorem 4.1, the case of the quadrics -- i.e. \(\prod_{i=1}^s d_i= 2\) -- is considered, obtaining that \(F_k(X)\) is empty if \(k>\lfloor\frac{n-s}{2} \rfloor\) and if \(k\le \lfloor\frac{n-s}{2} \rfloor\) then \(F_k(X)\) has one or two components, which are rationally connected.
Reviewer: Pietro De Poi (Udine)Abundance for 3-folds with non-trivial Albanese maps in positive characteristic.https://www.zbmath.org/1455.140322021-03-30T15:24:00+00:00"Zhang, Lei"https://www.zbmath.org/authors/?q=ai:zhang.lei.25|zhang.lei.13|zhang.lei.4|zhang.lei.1|zhang.lei.9|zhang.lei.5|zhang.lei.21|zhang.lei.15|zhang.lei.23|zhang.lei.20|zhang.lei.10|zhang.lei.11|zhang.lei.2|zhang.lei.7|zhang.lei.22|zhang.lei.18|zhang.lei.16|zhang.lei.24|zhang.lei.14|zhang.lei.17|zhang.leiIn recent years, many of the results of the Minimal Model Program have been extended from characteristic zero to characteristic \(p>5\). For example, existence of log minimal models of threefolds has been proved
by \textit{C. Birkar} [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 1, 169--212 (2016; Zbl 1346.14040)], \textit{C. D. Hacon} and \textit{C. Xu} [J. Am. Math. Soc. 28, No. 3, 711--744 (2015; Zbl 1326.14032)].
There has also been significant progress on the log abundance conjecture. In the case of threefold klt pairs,
it has been proved when the variety is of log general type or when the boundary divisor is big (in addition to the above references, see also
[\textit{P. Cascini} et al., Ann. Sci. Éc. Norm. Supér. (4) 48, No. 5, 1239--1272 (2015; Zbl 1408.14020)] and [\textit{C. Xu}, J. Inst. Math. Jussieu 14, No. 3, 577--588 (2015; Zbl 1346.14020)]).
In the present paper, the author builds upon his previous work [\textit{L. Zhang}, J. Lond. Math. Soc., II. Ser. 99, No. 2, 332--348 (2019; Zbl 1410.14013)] and proves abundance for threefolds
with non-trivial Albanese map.
Theorem 1.1. Let \(X\) be a klt, \(\mathbb{Q}\)-factorial, projective minimal threefold defined over an algebraically closed fields \(k\) of characteristic \(p>5\). Assume that the Albanese map is non-trivial. Then \(K_X\) is semi-ample.
The author also proves some instances of log abundance.
Theorem 1.2. Let \((X,B)\) be a klt, \(\mathbb{Q}\)-factorial, projective minimal pair of dimension three defined over an algebraically closed field of characteristic \(p>5\). Assume that the Albanese map \(\alpha_X\) is non-trivial.
Denote by \(f:X\rightarrow Y\) the fibration arising from the Stein factorization of \(\alpha_X\) and by \(X_{\eta}\) the generic fiber of \(f\). Assume moreover that \(B=0\) if
(1) \(\operatorname{dim}(Y)=2\) and \(\kappa(X_{\eta}, (K_X+B)|_{X_{\eta}})=0\), or
(2) \(\operatorname{dim}(Y)=1\) and \(\kappa(X_{\eta}, (K_X+B)|_{X_{\eta}})=1\).
Then \(K_X+B\) is semi-ample.
Reviewer: Justin Lacini (Lawrence)Spherical indecomposable representations of Lie superalgebras.https://www.zbmath.org/1455.141002021-03-30T15:24:00+00:00"Sherman, Alexander"https://www.zbmath.org/authors/?q=ai:sherman.alexanderFor a quasireductive Lie superalgebra \(\mathfrak{g}\) and representation \(\rho:\mathfrak{g}\to\mathfrak{g}\mathfrak{l}(V)\),
the triple \((V,\mathfrak{g},\rho)\) is spherical if there exists a Borel subalgebra \(\mathfrak{b}\) of \(\mathfrak{g}\) and a vector \(v\in V_{\overline{0}}\) such that \((\rho(\mathfrak{b})+\mathbb{C}\cdot \text{id}_{V})v=V\).
Assuming \(\mathfrak{g}\) is one of the following Lie superalgebras \(\mathfrak{gl}(m|n)\), \(\mathfrak{osp}(m|2n)\), \(\mathfrak{p}(n)\), \(\mathfrak{q}(n)\), or exceptional basic simple over the complex numbers, A. Sherman describes all spherical indecomposable representations of \(\mathfrak{g}\), i.e., all finite-dimensional indecomposable \(\mathfrak{g}\)-modules \(V\) such that there exists an even vector \(v\in V_{\overline{0}}\) and a Borel subalgebra \(\mathfrak{b}\subseteq \mathfrak{g}\) such that \((\mathfrak{b}+\mathbb{C}\cdot\text{id}_V) v=V\).
That is, he classifies such triples \((V,\mathfrak{g},\rho)\) up to equivalence,
where \((V,\mathfrak{g},\rho)\sim(V',\mathfrak{g}',\rho')\) if and only if \(V\cong V'\) and \(\rho(\mathfrak{g})+\mathbb{C}\cdot \text{id}_{V}\) maps to \(\rho'(\mathfrak{g}')+\mathbb{C}\cdot \text{id}_{V'}\) under the induced isomorphism \(\mathfrak{g}\mathfrak{l}(V)\to\mathfrak{g}\mathfrak{l}(V')\). Here, the author only considers the cases when the odd superspace \(V_{\overline{1}}\) is nonzero, or else, he is considering the action of a reductive group, which is already known. Thus, Sherman extends Kac's result, who classified all visible linear groups acting irreducibly on a vector space, thus solving the problem of finding all spherical irreducible representations of reductive Lie algebras.
Reviewer: Mee Seong Im (West Point)Higher spin \(\mathfrak{sl}_2R\)-matrix from equivariant (co)homology.https://www.zbmath.org/1455.580132021-03-30T15:24:00+00:00"Bykov, Dmitri"https://www.zbmath.org/authors/?q=ai:bykov.dmitri-v"Zinn-Justin, Paul"https://www.zbmath.org/authors/?q=ai:zinn-justin.paulSummary: We compute the rational \(\mathfrak{sl}_2 R\)-matrix acting in the product of two spin-\(\frac{\ell}{2}\) (\(\ell \in\mathbb{N}\)) representations, using a method analogous to the one of \textit{D. Maulik} and \textit{A. Okounkov} [Quantum groups and quantum cohomology. Paris: Société Mathématique de France (SMF) (2019; Zbl 1422.14002)], i.e., by studying the equivariant (co)homology of certain algebraic varieties. These varieties, first considered by \textit{N. Nekrasov} and \textit{S. Shatashvili} [AIP Conf. Proc. 1134, 154--169 (2009; Zbl 1180.81125)], are typically singular. They may be thought of as the higher spin generalizations of \(A_1\) Nakajima quiver varieties (i.e., cotangent bundles of Grassmannians), the latter corresponding to \(\ell =1\).Boundedness properties of automorphism groups of forms of flag varieties.https://www.zbmath.org/1455.140812021-03-30T15:24:00+00:00"Guld, A."https://www.zbmath.org/authors/?q=ai:guld.aSummary: We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.)
Let \(K\) be a field of characteristic 0, containing all roots of unity. Let the \(K\)-variety \(X\) be a form of an admissible flag variety. We prove that \(X\) is either ruled, or the automorphism group of \(X\) is bounded, meaning that there exists a constant \(C \in \mathbb{N}\) such that if \(G\) is a finite subgroup of \(\Aut_K(X)\), then the cardinality of \(G\) is smaller than \(C\).On the relative twist formula of \(\ell\)-adic sheaves.https://www.zbmath.org/1455.140402021-03-30T15:24:00+00:00"Yang, En Lin"https://www.zbmath.org/authors/?q=ai:yang.enlin"Zhao, Yi Geng"https://www.zbmath.org/authors/?q=ai:zhao.yigengSummary: We propose a conjecture on the relative twist formula of \(\ell\)-adic sheaves, which can be viewed as a generalization of Kato-Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.On determinantal ideals and algebraic dependence.https://www.zbmath.org/1455.130032021-03-30T15:24:00+00:00"Barile, Margherita"https://www.zbmath.org/authors/?q=ai:barile.margherita"Macchia, Antonio"https://www.zbmath.org/authors/?q=ai:macchia.antonioSummary: Let \(X\) be a matrix with entries in a polynomial ring over an algebraically closed field \(K\). We prove that, if the entries of \(X\) outside some \((t\times t)\)-submatrix are algebraically dependent over \(K\), the arithmetical rank of the ideal \(l_t(X)\) of \(t\)-minors of \(X\) drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by \(k\) if \(X\) has \(k\) zero entries. This upper bound turns out to be sharp if \(\mathrm{char}\, K=0\), since it then coincides with the lower bound provided by the local cohomological dimension.Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings.https://www.zbmath.org/1455.130302021-03-30T15:24:00+00:00"Reichelt, Thomas"https://www.zbmath.org/authors/?q=ai:reichelt.thomas"Saito, Morihiko"https://www.zbmath.org/authors/?q=ai:saito.morihiko"Walther, Uli"https://www.zbmath.org/authors/?q=ai:walther.uliLet \(X\) be a projective scheme over \(\mathbb{C}\) with a very ample line bundle \(\mathcal{L}\), and let \(\mathcal{C}\) be the cone of \(X\) associated with \(\mathcal{L}\). Let \(x_{1},\dots,x_{n}\) be the projective coordinates of \(\mathbb{P}^{n-1}(\mathbb{C})\) such that \(X\subseteq \mathbb{P}^{n-1}(\mathbb{C})\) and \(\mathcal{O}_{\mathbb{P}^{n-1}(\mathbb{C})}(1)\vert_{X}=\mathcal{L}\), and let \(\mathfrak{a} \subseteq R=\mathbb{C}[x_{1},\dots,x_{n}]\) be the defining ideal of \(\mathcal{C}\subseteq \mathbb{A}^{n}_{\mathbb{C}}\). Then the Lyubeznik numbers of \(\mathcal{C}\) are defined as
\[\lambda_{i,j}(\mathcal{C}):= \dim_{\mathbb{C}} \mathrm{Ext}_{R}^{i}\left(\mathbb{C},H_{\mathfrak{a}}^{n-j}(R)\right)\]
for every \(i,j \geq 0\). The authors specify a technical condition that implies the dependence of the Lyubeznik numbers \(\lambda_{i,j}(\mathcal{C})\) on the choice of \(\mathcal{L}\). Using this result, they prove that if \(k\) is a field of characteristic \(0\), then there exist projective schemes over \(k\) such that the Lyubeznik numbers \(\lambda_{i,j}(\mathcal{C})\) of their cone \(\mathcal{C}\) depend on their projective embeddings. This answers a question of [\textit{G. Lyubeznik}, Lect. Notes Pure Appl. Math. 226, 121--154 (2002; Zbl 1061.14005), p. 133] negatively. The striking feature of this result is that it contrasts entirely the positive characteristic case where the machinery of Frobenius endomorphism is available; see [\textit{L. Núñez-Betancourt} et al., Contemp. Math. 657, 137--163 (2016; Zbl 1346.13035)] and [\textit{W. Zhang}, Adv. Math. 228, No. 1, 575--616 (2011; Zbl 1235.13012)].
Reviewer: Hossein Faridian (Clemson)Integral points on the elliptic curve \(E_{ pq }\): \(y^2 = x^3 + ( pq - 12) x - 2( pq - 8)\).https://www.zbmath.org/1455.110822021-03-30T15:24:00+00:00"Cheng, Teng"https://www.zbmath.org/authors/?q=ai:cheng.teng"Ji, Qingzhong"https://www.zbmath.org/authors/?q=ai:ji.qingzhong"Qin, Hourong"https://www.zbmath.org/authors/?q=ai:qin.hourongSummary: Let \(p = 8k + 5, q = 8k + 3\) be the twin prime pair for some nonnegative integer \(k\). Assume that \(\left(\frac{5}{p} \right) = - 1\) or \(\left(\frac{7}{q} \right) = - 1\). In this paper, we prove that the elliptic curve \(E_{ pq }\): \(y^2 = x^3 + ( pq - 12)x - 2( pq - 8)\) has unique integral point (2, 0).More variants of Erdős-Selfridge superelliptic curves and their rational points.https://www.zbmath.org/1455.110912021-03-30T15:24:00+00:00"Saradha, N."https://www.zbmath.org/authors/?q=ai:saradha.nSummary: Developing on the works of \textit{M. A. Bennett} and \textit{S. Siksek} [Compos. Math. 152, No. 11, 2249--2254 (2016; Zbl 1407.11081)] and more recently of \textit{P. Das} et al. [Mathematika 64, No. 2, 380--386 (2018; Zbl 1439.11151)], we study rational points on several other variants of Erdős-Selfridge super elliptic curve.Automorphisms of \(K3\) surfaces and their applications.https://www.zbmath.org/1455.140772021-03-30T15:24:00+00:00"Taki, Shingo"https://www.zbmath.org/authors/?q=ai:taki.shingoSummary: This paper is a survey about \(K3\) surfaces with an automorphism and log rational surfaces, in particular, log del Pezzo surfaces and log Enriques surfaces. It is also a reproduction on my talk at ``Mathematical structures of integrable systems and their applications'' held at Research Institute for Mathematical Sciences in September 2018.Periodic and quasi-periodic solutions of Toda lattice via hyperelliptic \(\sigma\) functions.https://www.zbmath.org/1455.140662021-03-30T15:24:00+00:00"Matsutani, Shigeki"https://www.zbmath.org/authors/?q=ai:matsutani.shigekiSummary: In this report, I summarize results in the paper \textit{Y. Kodama} et al. [Ann. Inst. Fourier 63, No. 2, 655--688 (2013; Zbl 1279.14044)] to pose a problem to give an explicit relation between periodic and quasi-periodic solutions of Toda lattice. For a hyperelliptic curve \(X_g\) of genus \(g\), we have a quasi-periodic solution of Toda lattice in terms of the hyperelliptic \(\sigma\) function and its addition theorem. Using the division polynomial of \(X_g\), we find \(2N\)-division points in its Jacobi variety and then have \(N\)-periodic solution of Toda-lattice. It is well-known that the \(N\)-periodic solution is associated with a hyperellptic curve \(\widehat{X}_{g,N-1}\) of genus \(N-1\) rather than \(g\). However it is not clear how \(X_g\) and \(\widehat{X}_{g,N-1}\) are connected geometrically, though the problem is very simple and natural. In this report, after I give a review of the recent development of \(\sigma\) function theory of higher genus and show the summary of our previous work, I give some comments on the problem.Quiver bundles and wall crossing for chains.https://www.zbmath.org/1455.140682021-03-30T15:24:00+00:00"Gothen, P. B."https://www.zbmath.org/authors/?q=ai:gothen.peter-b"Nozad, A."https://www.zbmath.org/authors/?q=ai:nozad.azizehA holomorphic \((m +1)\)-chain on a compact Riemann surface \(X\) of genus \(g\) is a diagram
\[ E_m \stackrel{\phi_m}{\rightarrow} E_{m-1} \stackrel{\phi_{m-1}}{\rightarrow} \cdots
\stackrel{\phi_2}{\rightarrow} E_1 \stackrel{\phi_1}{\rightarrow} E_0,\]
where each \(E_i\) is a holomorphic vector bundle and \(\phi_i \colon E_i \rightarrow E_{i-1}\) is a holomorphic map.
Moduli spaces of holomorphic chains depend on a stability parameter \(\alpha= (\alpha_0, \ldots, \alpha_m)\).
Using the the Hitchin--Kobayashi correspondence for quiver bundles,
the authors present a new algebraic proof of the following result
of Álvarez-Cónsul, García-Prada, and Schmitt [\textit{L. Álvarez-Cónsul} et al., IMRP, Int. Math. Res. Pap. 2006, No. 10, Article ID 73597, 82 p. (2006; Zbl 1111.32012), Proposition 4.4]:
Let \(C'\) and \(C''\) be \(\alpha\)-polystable
holomorphic chains and let
\(\alpha_i-\alpha_{i-1} \geq 2g-2\) for all \(i = 1,..., m\).
Then the following inequalities hold:
\[\mu(\ker(d)) \leq \mu_\alpha(C')- \mu_\alpha(C'')\]
\[ \mu (\mbox{coker}(d)) \geq \mu_\alpha (C')- \mu_\alpha(C'') + 2g-2 \]
where \(d \colon \mathcal H^0 \rightarrow \mathcal H^1\) is given by
\(d(g_0,..., g_m) = (g_{i-1} \circ \phi''_i- \phi'_i \circ g_i)\) for \(g_i \in \mbox{Hom}(E''_i , E'_ i).\)
Reviewer: Elizabeth Gasparim (Antofagasta)Recursion relations on the power series expansion of the universal Weierstrass sigma function.https://www.zbmath.org/1455.330112021-03-30T15:24:00+00:00"Eilbeck, J. Chris"https://www.zbmath.org/authors/?q=ai:eilbeck.john-chris"Ônishi, Yoshihiro"https://www.zbmath.org/authors/?q=ai:onishi.yoshihiroSummary: The main aim of this paper is an exposition of the theory of Buchstaber and Leykin on the heat equations for the multivariate sigma functions. We treat only the elliptic curve case, but keeping the most general elliptic curve equation, which may be useful for number theoretic applications.Special cubic birational transformations of projective spaces.https://www.zbmath.org/1455.140262021-03-30T15:24:00+00:00"Staglianò, Giovanni"https://www.zbmath.org/authors/?q=ai:stagliano.giovanniBirational transformations from a projective space \(\mathbb P^n\) are called special when the base locus is smooth and connected. Many articles have been devoted to the problem of classifying special birational transformations whose base locus has dimension at most three.
In this article, the author classifies birational transformations \(\mathbb P^6\dashrightarrow Z\) that are defined by cubic polynomials, where \(Z\) is a locally factorial complete intersection. As a consequence he completes the classification of all special birational transformations into a factorial complete intersection whose base locus has dimension at most three. There are 28 types such that the inverse map is not linear, and 9 types where it is linear. Many examples are given.
The article also contains a survey of recent results and a rich bibliography about this classical and fascinating topic.
Reviewer: Emilia Mezzetti (Trieste)Structural theorem for \(gr\)-injective modules over \(gr\)-Noetherian \(G\)-graded commutative rings and local cohomology functors.https://www.zbmath.org/1455.130292021-03-30T15:24:00+00:00"Lu, Li"https://www.zbmath.org/authors/?q=ai:lu.li.1|lu.liThe author studies graded modules over graded commutative rings in analogy to the classical theory. He introduces and studies gr-Bass numbers for gr-noetherian modules over gr-noetherian graded rngs, and expresses them in terms of the functor \(Ext\). Further topics include radical and preradical functors, etc. The author also defines and uses abstract local cohomology functors.
Reviewer: Moshe Roitman (Haifa)Identifiability in phylogenetics using algebraic matroids.https://www.zbmath.org/1455.921032021-03-30T15:24:00+00:00"Hollering, Benjamin"https://www.zbmath.org/authors/?q=ai:hollering.benjamin"Sullivant, Seth"https://www.zbmath.org/authors/?q=ai:sullivant.sethSummary: Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters are generically identifiable for 2-tree CFN and K3P mixtures. We also show that the \(k\)-cycle phylogenetic network parameter is identifiable under the K2P and K3P models.Rational sequences on different models of elliptic curves.https://www.zbmath.org/1455.110492021-03-30T15:24:00+00:00"Çelik, Gamze Savaş"https://www.zbmath.org/authors/?q=ai:savas-celik.gamze"Sadek, Mohammad"https://www.zbmath.org/authors/?q=ai:sadek.mohammad"Soydan, Gökhan"https://www.zbmath.org/authors/?q=ai:soydan.gokhanLet \(K\) be a number field and \(S=\{s_{1},\ldots, s_{k}\}\subset K\). The authors are interested in finding a polynomial \(f\in K[x,y]\) of certain form defining genus one curve \(C:\;f(x,y)=0\) with the following property: for each \(i\in\{1, \ldots, k\}\) there is \(y_{i}\in S\) such that the point \(P_{i}=(s_{i}, y_{i})\) lies on the curve \(C\), i.e., \(P_{i}\) is a \(K\)-rational point on \(C\). We will say that \(S\) is realisable by \(f\).
Several results are proved in the paper. For example, if \(S=\{-1, 0, 1, s_{2}, s_{3}, s_{4}\}\) then (under mild conditions on \(s_{i}\)) there are infinitely many values of \(d\) such that the set \(S\) is realisable by the polynomial \(F_{d}(x,y)=x^2+y^2-1-dx^2y^2\). Note that the curve (of genus 1) defined by \(F_{d}\) is so called Edwards curve (over \(k\)). Related result is proved for twisted Edwards curves. Moreover, a similar result is proved for polynomials \(F_{a,b}=ax(y^2-1)-by(x^2-1)\), which define so called Huff curve. The proofs are elementary and are based on simple manipulations of polynomial equations defining objects under consideration.
Reviewer: Maciej Ulas (Kraków)Degrees of iterates of rational maps on normal projective varieties.https://www.zbmath.org/1455.140282021-03-30T15:24:00+00:00"Dang, Nguyen-Bac"https://www.zbmath.org/authors/?q=ai:dang.nguyen-bacSummary: Let \(X\) be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of \(X\), recovering former results by
\textit{T.-C. Dinh} and \textit{N. Sibony} [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 959--971 (2004; Zbl 1074.53058)], and by
\textit{T. T. Truong} [J. Reine Angew. Math. 758, 139--182 (2020; Zbl 07148358)]. Precisely, we give a new proof of the submultiplicativity properties of these degrees and of their birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by
\textit{J. Xiao} [Ann. Inst. Fourier 65, No. 3, 1367--1379 (2015; Zbl 1333.32025)] and
\textit{D. Popovici} [Math. Ann. 364, No. 1--2, 649--655 (2016; Zbl 1341.32015)], which generalizes
Siu's inequality (see [\textit{S. Trapani}, Math. Z. 219, No. 3, 387--401 (1995; Zbl 0828.14002)] to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in \(X\).The containment poset of type \(A\) Hessenberg varieties.https://www.zbmath.org/1455.140962021-03-30T15:24:00+00:00"Drellich, E."https://www.zbmath.org/authors/?q=ai:drellich.elizabethSummary: Flag varieties are well-known algebraic varieties with many important geometric, combinatorial, and representation theoretic properties. A Hessenberg variety is a subvariety of a flag variety identified by two parameters: an element \(X\) of the Lie algebra \(\mathfrak{g}\) and a Hessenberg subspace \(H\subseteq \mathfrak{g} \). This paper considers when two Hessenberg spaces define the same Hessenberg variety when paired with \(X\). To answer this question we present the containment poset \(\mathcal{P}_X\) of type \(A\) Hessenberg varieties with a fixed first parameter \(X\) and give a simple and elegant proof that if \(X\) is not a multiple of the element \(\mathbf{1}\) then the Hessenberg spaces containing the Borel subalgebra determine distinct Hessenberg varieties. Lastly we give a natural involution on \(\mathcal{P}_X\) that induces a homeomorphism of varieties and prove additional properties of \(\mathcal{P}_X\) when \(X\) is a regular nilpotent element.Smooth deformations of singular contractions of class VII surfaces.https://www.zbmath.org/1455.140732021-03-30T15:24:00+00:00"Dloussky, Georges"https://www.zbmath.org/authors/?q=ai:dloussky.georges"Teleman, Andrei"https://www.zbmath.org/authors/?q=ai:teleman.andrei-dumitruThe authors study normal compact surfaces \(Y\) obtained from a minimal class VII surface \(X\) by contraction of a cycle \(C\) of \(r\) rational curves with \(C^2 < 0\). The main result states that, if the obtained cusp is smoothable, then \(Y\) is globally smoothable. The proof relies on a vanishing theorem for \(H^2(\Theta_Y )\). The condition ``the cusp is smoothable'' in the main theorem can be checked in terms of the intersection numbers of the cycle, by the Looijenga conjecture, which has recently been proved by \textit{M. Gross} et al. [Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)]. The authors also prove that this condition is always satisfied if \(r < b_2(X)\leq 11\). Therefore the singular surface \(Y\) aforementioned with \(r < b2(X)\leq 11\) is always smoothable by rational surfaces and this holds even for
unknown class VII surfaces. Furthermore, the authors show that if \(r < b_2(X)\), any smooth small deformation of \(Y\) is rational, and if \(r = b_2(X)\) (i.e. when \(X\) is a half-Inoue surface), any smooth small deformation of \(Y\) is an Enriques surface.
Reviewer: Quanting Zhao (Wuhan)On the acceptable elements.https://www.zbmath.org/1455.140882021-03-30T15:24:00+00:00"He, Xuhua"https://www.zbmath.org/authors/?q=ai:he.xuhua"Nie, Sian"https://www.zbmath.org/authors/?q=ai:nie.sianSummary: In this article, we study the set \(B(G, \{\mu\})\) of acceptable elements for any \(p\)-adic group \(G\). We show that \(B(G, \{\mu\})\) contains a unique maximal element and the maximal element is represented by an element in the admissible subset of the associated Iwahori-Weyl group.Towards an intersection Chow cohomology theory for GIT quotients.https://www.zbmath.org/1455.140902021-03-30T15:24:00+00:00"Edidin, Dan"https://www.zbmath.org/authors/?q=ai:edidin.dan"Satriano, Matthew"https://www.zbmath.org/authors/?q=ai:satriano.matthewSummary: We study the Fulton-MacPherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with \(\mathbb{Q} \)-coefficients, every operational class can be represented by a \textit{topologically strong} cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Our methods also allow us to prove that if \(X\) is the good moduli space of a properly stable, smooth, Artin stack then the natural map \(\operatorname{Pic}(X)_\mathbb{Q} \), \(L \mapsto c_1(L)\) is an isomorphism.Approximation by piecewise-regular maps.https://www.zbmath.org/1455.141092021-03-30T15:24:00+00:00"Bilski, Marcin"https://www.zbmath.org/authors/?q=ai:bilski.marcin"Kucharz, Wojciech"https://www.zbmath.org/authors/?q=ai:kucharz.wojciechSummary: A real algebraic variety \(W\) of dimension \(m\) is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of \(\mathbb{R}^m\). Let \(l\) be any nonnegative integer. We prove that every map of class \(\mathcal{C}^l\) from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the \(\mathcal{C}^l\) topology by piecewise-regular maps of class \(\mathcal{C}^k\), where \(k\) is an arbitrary integer satisfying \(k \geq l\). Next we derive consequences regarding algebraization of topological vector bundles.Completing perfect complexes. With appendices by Tobias Barthel and Bernhard Keller.https://www.zbmath.org/1455.180092021-03-30T15:24:00+00:00"Krause, Henning"https://www.zbmath.org/authors/?q=ai:krause.henningThe author introduces the notion of sequential completion of a category, which is the categorical analogue of the construction of the real numbers from the rationals via Cauchy sequences. The author gives conditions under which the completion of an abelian (resp. triangulated) category is again abelian (resp. triangulated) and shows that the theories of completion of abelian categories and derived categories are compatible to one another.
The paper contains numerous explicit examples of completions of categories. For a noetherian algebra \(\Lambda\) over a complete local ring, it is shown that the completion of the category of finite length \(\Lambda\)-modules identifies with the category of artinian \(\Lambda\)-modules. The completion of the category of perfect complexes over a right coherent ring identifies with the bouded derived category of finitely presented modules. This holds for non-affine noetherian schemes as well and one can use this to give a direct construction of the singularity category.
This paper contains three appendices. ``The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra''. ``The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. It is shown that the objects are precisely the colimits of Cauchy sequences of perfect complexes that satisfy an intrinsic boundeness condition. In the final appendix, Bernard Keller introduces the notion of a morphic enhancement of a triangulated category and provides a foundation for completing a triangulated category.''
Reviewer: Luca Pol (Regensburg)Random matrix theory with an external source.https://www.zbmath.org/1455.600022021-03-30T15:24:00+00:00"Brézin, Edouard"https://www.zbmath.org/authors/?q=ai:brezin.edouard"Hikami, Shinobu"https://www.zbmath.org/authors/?q=ai:hikami.shinobuFrom publisher's description: ``This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities. We consider Gaussian random matrix models in the presence of a deterministic matrix source. In such models the correlation functions are known exactly for an arbitrary source and for any size of the matrices. The freedom given by the external source allows for various tunings to different classes of universality. The main interest is to use this freedom to compute various topological invariants for surfaces such as the intersection numbers for curves drawn on a surface of given genus with marked points, Euler characteristics, and the Gromov-Witten invariants. A remarkable duality for the average of characteristic polynomials is essential for obtaining such topological invariants. The analysis is extended to nonorientable surfaces and to surfaces with boundaries''.
This very large book is structured into a preface, acknowledgements, about this book, contents, 10 chapters (divided in 28 subchapters), references, index:
Chapter 1. Introduction -- Chapter 2. Gaussian means -- Chapter 3. External source -- Chapter 4. Characteristic polynomials and duality -- Chapter 5. Universality -- Chapter 6. Intersection numbers of curves -- Chapter 7. Intersection numbers of \(p\)-spin curves -- Chapter 8. Open intersection numbers -- Chapter 9. Non-orientable surfaces from Lie algebras -- Chapter 10. Gromov-Witten invariants, \(P^1\) model.
From about this book: ``We have presented most results in several earlier publications, but here we attempted to present a unified and concise exposition''. The book contains 139 references and the index about 34 items.
This book can be recommended for all readers, who are interested in this field.
Reviewer: Ludwig Paditz (Dresden)Logarithmic compactification of the Abel-Jacobi section.https://www.zbmath.org/1455.140212021-03-30T15:24:00+00:00"Marcus, Steffen"https://www.zbmath.org/authors/?q=ai:marcus.steffen"Wise, Jonathan"https://www.zbmath.org/authors/?q=ai:wise.jonathanSummary: Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. We also describe the attendant deformation theory and virtual fundamental class of this moduli space. This recovers the double ramification cycle, as well as variants associated to differentials.An explicit formula of the normalized Mumford form.https://www.zbmath.org/1455.140572021-03-30T15:24:00+00:00"Ichikawa, Takashi"https://www.zbmath.org/authors/?q=ai:ichikawa.takashiSummary: We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented as an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. Furthermore, this formula gives a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves. Therefore, one can describe the behavior of this form and hence of the Polyakov string measure around the Deligne-Mumford boundary.On Hecke's decomposition of the regular differentials on the modular curve of prime level.https://www.zbmath.org/1455.110832021-03-30T15:24:00+00:00"Gross, Benedict H."https://www.zbmath.org/authors/?q=ai:gross.benedict-hLet \(p\) be a prime number and \(\mathbb F_p\) the prime field of \(p\) elements. The class number of the field \(K=\mathbb Q(\sqrt{-p})\) is denoted by \(h\). The group \(G=\mathrm{SL}_2(\mathbb F_p)/\{\pm 1\}\) acts on the modular curve \(X\) associated with the principal congruence subgroup \(\Gamma(p)\). Erich Hecke decomposed the space \(H^0(X,\Omega^1)\) of regular differentials of \(X\) in the irreducible subspaces under the action of \(G\). Henceforth, let \(p\equiv 3 \mod 4, p>3\). Hecke found the \(G\)-invariant subspace \(V_0\) of \(H^0(X,\Omega^1)\) of dimension \(h(p-1)/2\) isomorphic to the \(h\)- copies of \(W\), where \(W\) is one of two irreducible representations of \(G\) of dimension \((p-1)/2\).
Further, he identified certain periods of these differentials as the periods of elliptic curves with complex multiplication by \(K\). On the other hand, the author has obtained an elliptic curve \(A(p)\) defined over the Hilbert class field \(H\) of \(K\) as a factor of the Jacobian of the modular curve \(X_0(p^2)\), and also
\textit{G. Shimura} [J. Math. Soc. Japan 25, 523--544 (1973; Zbl 0266.14017)] has obtained an abelian variety \(B(p)\) of dimension \(h\) as a simple factor of the same Jacobian.
In this article, the author proves the above Hecke's results using the character theory of \(G\), the Lefshetz fixed point formula, and the holomorphic fixed point formula. He explains how \(A(p)\) and \(B(p)\) relate to Hecke's distinguished subspace \(V_0\).
Let \(\mathcal{U}\) be an irreducible representation of \(G\) appeared in \(H^0(X,\Omega^1)\) of multiplicity \(m(\mathcal{U})\) and \(\mathcal{U}^\vee\) the dual representation of \(\mathcal{U}\). Since the singular cohomology \(H^1(X)\) is \(G\)-isomorphic to deRham cohomology \(H^1(X,\mathbb C)\) which is the extension of \(H^0(C,\Omega^1)\) by \(H^1(X,\mathcal{O})=H^0(X,\Omega^1)^\vee\), one can compute the sum of the multiplicities \(m(\mathcal{U})+ m(\mathcal{U^\vee})\) by the Lefshetz fixed point formula. Further, the result that \(m(W)-m(W^\vee)=h\) is proved by the holomorphic trace formula. This implies that \(H^0(X,\Omega^1)\) contains \(h\) copies of \(W\). Let \(PX(p)\) be the Shimura variety which is the coarse moduli space of generalized elliptic curves with a full level \(p\) structure, up to scaling. Let \(Y\) be the compactification of \(\Gamma(p)\backslash \mathfrak{H}^-\),where \(\mathfrak{H}^-\) is the lower half plane of \(\mathbb C\).
The group \(\mathrm{PGL}_2(\mathbb F_p)\) acts on \(PX(p)\) over \(\mathbb Q\). The curve \(PX(p)(\mathbb C)\) has two components \(X\) and \(Y\) and \(H^0(Y,\Omega^1)=H^0(X,\Omega^1)^\vee\). Further, the Hecke operator \(T_\ell (\ell:\text{prime}\ne p)\) acts on \(PX(p)\) and \(H^0(PX(p),\Omega^1)\). The action of \(T_\ell\) commutes with that of \(\mathrm{PGL}_2(\mathbb F_p)\). Let \(\mathbb T\) be the commutative algebra over \(\mathbb Q\) generated by \(T_\ell\). The author constructs a CM-field \(E\) over \(K\) of degree \(h\), a Hecke character \(\chi:{\mathbb A}_K ^\times\rightarrow E^*\) and a surjective homomorphism of \(\mathbb T\) to the maximal real subfield \(E^+\) of \(E\).
Let \(W'\) be the conjugate representation of \(W\). Since \(W+W'\) is extended to the representation \(R\) of \(\mathrm{PGL}_2(\mathbb F_p)\), \(V_0\) gives a distinguished subspace \(V\) of dimension \(h(p-1)\) in \(H^0(PX(p),\Omega^1)\) over \(\mathbb Q\). Let \(f\) be the new form of weight \(2\) with coefficients in \(E^+\) with respect to \(\Gamma(p^2)\) determined by \(\chi\). Let \(M(E^+)\) be the subspace of dimension \(1\) over \(E^+\) of \(H^0(X_0(p^2),\Omega^1)\) spanned by all conjugates of \(f\) over \(\mathbb Q\). Since \(X_0(p^2)\) is the quotient of \(PX(p)\) by the action of a split torus in \(\mathrm{PGL}_2(\mathbb F_p)\), \(M(E^+)\) is realized as a subspace of \(H^0(PX(p),\Omega^1)\),which is denoted by the same notation.
The author shows that \(V\) is isomorphic to the simple module \(M(E^+)\otimes R\) over \(\mathbb Q\) under the action of \(\mathbb T\times\mathbb Q[\mathrm{PGL}_2(\mathbb F_p)]\) (See Theorem 8).
In particular \(V_0\) is isomorphic to \(M(E^+)\otimes W\).
Let \(\rho_A\) be the Hecke character of \(H\) obtained as the composition of \(\chi\) and the norm map from \(H\) to \(K\). Then \(\rho_A\) determines an isogeny class of elliptic curves defined over \(H\) with complex multiplication by the integers of \(K\). The elliptic curve \(A(p)\) is defined as the elliptic curve of the minimal discriminant \(-p^3\) in this class and \(B(p)\) is the abelian variety associated with the module \(M(E^+)\) by Shimura theory, and \(B(p)\) is obtained from \(A(p)\) by restriction of scalars. In the last place, the author gives a summary of what is known about the arithmetic of \(A(p)\) and \(B(p)\).
Reviewer: Noburo Ishii (Kyoto)The Hensel-Shafarevich canonical basis in Honda formal modules.https://www.zbmath.org/1455.111572021-03-30T15:24:00+00:00"Vostokov, Sergeĭ Vladimirovich"https://www.zbmath.org/authors/?q=ai:vostokov.sergei-vladimirovich"Vostokova, Regina Petrovna"https://www.zbmath.org/authors/?q=ai:vostokova.regina-petrovna"Ikonnikova, Elena Valer'evna"https://www.zbmath.org/authors/?q=ai:ikonnikova.elena-valerevnaSummary: In this paper, we construct Hensel-Shafarevich generating set in Honda formal modules over a higher dimensional field. Later, that should allow us to compute Hilbert symbol in this case.Exact WKB and abelianization for the \(T_3\) equation.https://www.zbmath.org/1455.810252021-03-30T15:24:00+00:00"Hollands, Lotte"https://www.zbmath.org/authors/?q=ai:hollands.lotte"Neitzke, Andrew"https://www.zbmath.org/authors/?q=ai:neitzke.andrewThe authors describe the exact WKB method from the viewpoint of abelenization , for Schrödinger operators as well as for their higher- order analogues. The authors find a new example, which is defined as ``\(T_3\) equation'' which is in fact a trice- punctured sphere, with regular singularities at the punctures.
In this case, one show, the exact WKB analysis leads for consideration of a new sort of Darboux coordinate system on a moduli space of flat SL(3) connections. The authors give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the \(T_3\) equation has the expected asymptotic properties. One also briefly revisit the Schrodinger equation equation with cubic potential and the Mathieu equation from the point of view of abelenization.
Reviewer: Alex B. Gaina (Chisinau)Convex lattice polygons with all lattice points visible.https://www.zbmath.org/1455.520132021-03-30T15:24:00+00:00"Morrison, Ralph"https://www.zbmath.org/authors/?q=ai:morrison.ralph"Tewari, Ayush Kumar"https://www.zbmath.org/authors/?q=ai:tewari.ayush-kumarSummary: Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than 2, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications.https://www.zbmath.org/1455.140542021-03-30T15:24:00+00:00"Xue, Cong"https://www.zbmath.org/authors/?q=ai:xue.congSummary: In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.On families of hyperelliptic curves over the field of rational numbers, whose Jacobian contains torsion points of given orders.https://www.zbmath.org/1455.110892021-03-30T15:24:00+00:00"Fedorov, Gleb Vladimirovich"https://www.zbmath.org/authors/?q=ai:fedorov.gleb-vladimirovichSummary: One of the pressing contemporary problems of algebra and number theory is the problem of the existence and searching for fundamental \(S\)-units in hyperelliptic fields. The problem of the existence and searching of \(S\)-units in hyperelliptic fields is equivalent the solvability of the norm equation -- the functional Pell equation -- with some additional conditions on the form of this equation and its solution. There is a deep connection between points of finite order in Jacobian variety (Jacobian) of hyperelliptic curve and nontrivial \(S\)-units of hyperelliptic field. This connection formed the basis of the algebraic approach proposed by V.P. Platonov to the well-known fundamental problem of boundedness of torsion in Jacobian varieties of hyperelliptic curves. For elliptic curves over a field of rational numbers, the torsion problem was solved by
\textit{B. Mazur} [Lect. Notes Math. 601, 107--148 (1977; Zbl 0357.14005)] in the 1970s. For curves of genus 2 and higher over the field of rational numbers, the torsion problem turned out to be much more complicated, and it is far from its complete solution. The main results obtained in this direction include to the description of torsion subgroups of Jacobian varieties of specific hyperelliptic curves, and also to the description of some families of hyperelliptic curves of the genus \(g \ge 2\).
In this article, we have found a new method for studying solvability. functional norm equations giving a full description hyperelliptic curves over the field of rational numbers, whose Jacobian varieties possess torsion points of given orders. Our method is based on an analytical study of representatives finite order divisors in a divisor class group of degree zero and their Mumford representations. As an illustration of the operation of our method in this article, we directly found all parametric families of hyperelliptic curves of genus two over the field of rational numbers, whose Jacobian varieties have rational torsion points of orders not exceeding five. Moreover, our method allows us to determine which parametric family found this curve belongs, whose Jacobian has a torsion point of order not exceeding five.Equivalence of K3 surfaces from Verra threefolds.https://www.zbmath.org/1455.140372021-03-30T15:24:00+00:00"Kapustka, Grzegorz"https://www.zbmath.org/authors/?q=ai:kapustka.grzegorz"Kapustka, Michał"https://www.zbmath.org/authors/?q=ai:kapustka.michal"Moschetti, Riccardo"https://www.zbmath.org/authors/?q=ai:moschetti.riccardoSummary: We study \((2,2)\) divisors in \(\mathbb{P}^2\times\mathbb{P}^2\) giving rise to pairs of nonisomorphic, derived equivalent, and \(\mathbb{L}\)-equivalent \(K3\) surfaces of degree \(2\). In particular, we confirm the existence of such fourfolds as predicted recently by \textit{A. Kuznetsov} and \textit{E. Shinder} [Sel. Math., New Ser. 24, No. 4, 3475--3500 (2018; Zbl 1450.11036)].Arbitrarily large 2-torsion in Tate-Shafarevich groups of abelian varieties.https://www.zbmath.org/1455.110902021-03-30T15:24:00+00:00"Flynn, E. V."https://www.zbmath.org/authors/?q=ai:flynn.eugene-victorIn [\textit{E. V. Flynn}, J. Number Theory 186, 248--258 (2018; Zbl 1444.11136)], it is shown that the Tate-Shafarevich groups of absolutely simple Jacobians of genus 2 curves over \(\mathbb Q\) (in particular, their 2-torsion) can be arbitrarily large. The paper under review generalizes this result. More precisely, it is proved that for any \(g \geq 1\), there exists a hyperelliptic curve of genus \(g\) over \(\mathbb Q\), with absolutely simple Jacobian, such that the 2-torsion part of the Tate-Shafarevich groups is arbitrarily large amongst its quadratic twists. An important ingredient of the proof is a recent construction that for any \(g\) describes curves of genus \(g\) whose Jacobians admit a \((2,\ldots, 2)\) isogeny [\textit{J.-F. Mestre}, J. Algebr. Geom. 22, No. 3, 575--580 (2013; Zbl 1312.14085)].
Reviewer: Dimitros Poulakis (Thessaloniki)Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties.https://www.zbmath.org/1455.140862021-03-30T15:24:00+00:00"Zhao, Heer"https://www.zbmath.org/authors/?q=ai:zhao.heerSummary: We show -- for a semistable abelian variety \(A_K\) over a complete discrete valuation field \(K\) -- that every finite-subgroup scheme of \[A_K\] extends to a log finite-flat group scheme over the valuation ring of \(K\) endowed with the canonical log structure. To achieve this, we first give a positive answer to a question of Nakayama, namely whether every weak log-abelian variety over an \textit{fs (fine and saturated)} log scheme with its underlying scheme locally noetherian is a sheaf for the Kummer-flat topology. We also give several equivalent conditions defining isogenies of log-abelian varieties.Analytical and number-theoretical properties of the two-dimensional sigma function.https://www.zbmath.org/1455.110372021-03-30T15:24:00+00:00"Ayano, Takanori"https://www.zbmath.org/authors/?q=ai:ayano.takanori"Bukhshtaber, Viktor Matveevich"https://www.zbmath.org/authors/?q=ai:bukhshtaber.viktor-matveevichSummary: This survey is devoted to the classical and modern problems related to the entire function \({\sigma({\mathbf{u}};\lambda)} \), defined by a family of nonsingular algebraic curves of genus 2, where \({\mathbf{u}} = (u_1,u_3)\) and \(\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})\). It is an analogue of the Weierstrass sigma function \(\sigma({{u}};g_2,g_3)\) of a family of elliptic curves. Logarithmic derivatives of order \(2\) and higher of the function \({\sigma({\mathbf{u}};\lambda)}\) generate fields of hyperelliptic functions of \({\mathbf{u}} = (u_1,u_3)\) on the Jacobians of curves with a fixed parameter vector \(\lambda \). We consider three Hurwitz series \(\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}, \sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}\) and \(\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!} \). The survey is devoted to the number-theoretic properties of the functions \(a_{m,n}(\lambda), \xi_k(u_1;\lambda)\) and \(\mu_k(u_3;\lambda)\). It includes the latest results, which proofs use the fundamental fact that the function \({\sigma ({\mathbf{u}};\lambda)}\) is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.Monodromy of rational curves on toric surfaces.https://www.zbmath.org/1455.141132021-03-30T15:24:00+00:00"Lang, Lionel"https://www.zbmath.org/authors/?q=ai:lang.lionelFor an ample line bundle \(\mathcal{L}\) on a complete toric surface \(X\), the author considers the subset \(V_{\mathcal{L}}\subset |\mathcal{L}|\) of irreducible, nodal, rational curves contained in the smooth locus of \(X\). For a general curve \(C\in V_{\mathcal{L}}\), any loop in \(V_{\mathcal{L}}\) based at \(C\) induces a permutation on the set of nodes of the curve \(C\). This action is recorded by the monodromy map. The author defines the obstruction map \(\Psi _{X}:\{\text{ nodes of }C\} \rightarrow (H_1(X^{\bullet},\mathbb{Z})/\pi (\mathcal{P}))/\pm\mathrm{id}\), where \(\mathcal{P}\) is the subspace generated by the punctures of \(C^{\bullet}:=C\cap X^{\bullet}\),
\(X^{\bullet}\) being the two-dimensional torus orbit in \(X\). He shows that the image of the monodromy is exactly the group of deck transformations of \(\Psi _{X}\), provided that \(\mathcal{L}\) is sufficiently big. He constructs a handy tool to compute the image of the monodromy for any pair \((X,\mathcal{L})\) and presents a family of pairs \((X,\mathcal{L})\) with small \(\mathcal{L}\) for which the image of the monodromy is strictly smaller than expected.
Reviewer: Vladimir P. Kostov (Nice)Fujiki relations and fibrations of irreducible symplectic varieties.https://www.zbmath.org/1455.140312021-03-30T15:24:00+00:00"Schwald, Martin"https://www.zbmath.org/authors/?q=ai:schwald.martinSummary: This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has rank \((3,0,b_2(X)-3)\). This enables us to
study fibrations of these varieties; imposing the newer definition from \textit{D. Greb} et al. [Adv. Stud. Pure Math. 70, 67--113 (2016; Zbl 1369.14052)] we show that they behave much like irreducible symplectic manifolds.Totally geodesic subvarieties in the moduli space of curves.https://www.zbmath.org/1455.140132021-03-30T15:24:00+00:00"Ghigi, Alessandro"https://www.zbmath.org/authors/?q=ai:ghigi.alessandro"Pirola, Gian Pietro"https://www.zbmath.org/authors/?q=ai:pirola.gian-pietro"Torelli, Sara"https://www.zbmath.org/authors/?q=ai:torelli.saraA note on the behaviour of the Tate conjecture under finitely generated field extensions.https://www.zbmath.org/1455.140472021-03-30T15:24:00+00:00"Ambrosi, Emiliano"https://www.zbmath.org/authors/?q=ai:ambrosi.emilianoSummary: We show that the \(\ell\)-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the \(\ell\)-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher codimension are given.Some specialization theorems for families of abelian varieties.https://www.zbmath.org/1455.140872021-03-30T15:24:00+00:00"Zannier, Umberto"https://www.zbmath.org/authors/?q=ai:zannier.umberto-mSummary: Consider an algebraic family \(\pi:\mathcal{A}\to B\) of abelian varieties, defined over \(\overline{\mathbb{Q}}\). We shall be concerned with properties of the generic fiber of \(\mathcal{A}\) which are preserved on restricting to some (or `many') suitable special fibers. We shall focus on instances like torsion for values of a section, endomorphism rings, existence of generic and special isogenies, illustrating some known results and some applications. Another, more recent, issue which we shall briefly discuss concerns the existence of abelian varieties over \(\overline{\mathbb{Q}}\) not isogenous to a Jacobian. We shall conclude with a few comments on other specialization issues.Unfolding of orbifold LG B-models: a case study.https://www.zbmath.org/1455.140792021-03-30T15:24:00+00:00"He, Weiqiang"https://www.zbmath.org/authors/?q=ai:he.weiqiang"Li, Si"https://www.zbmath.org/authors/?q=ai:li.si.2|li.si.1"Li, Yifan"https://www.zbmath.org/authors/?q=ai:li.yifanSummary: In this note we explore the variation of Hodge structures associated to the orbifold Landau-Ginzburg B-model whose superpotential has two variables. We extend the Getzler-Gauss-Manin connection to Hochschild chains twisted by group action. As an application, we provide explicit computations for the Getzler-Gauss-Manin connection on the universal (noncommutative) unfolding of \(\mathbb{Z}_2\)-orbifolding of A-type singularities. The result verifies an example of deformed version of McKay correspondence.Sheaf counting on local \(K3\) surfaces.https://www.zbmath.org/1455.141072021-03-30T15:24:00+00:00"Maulik, Davesh"https://www.zbmath.org/authors/?q=ai:maulik.davesh"Thomas, Richard P."https://www.zbmath.org/authors/?q=ai:thomas.richard-pSummary: There are two natural ways to count stable pairs or Joyce-Song pairs on \(X=K3\times\mathbb{C}\); one via weighted Euler characteristic and the other by virtual localisation of the reduced virtual class. Since \(X\) is noncompact these need not be the same. We show their generating series are related by an exponential.
As applications we prove two conjectures of Toda, and a conjecture of Tanaka-Thomas defining Vafa-Witten invariants in the semistable case.On the extended Hensel construction and its application to the computation of real limit points.https://www.zbmath.org/1455.130442021-03-30T15:24:00+00:00"Alvandi, Parisa"https://www.zbmath.org/authors/?q=ai:alvandi.parisa"Ataei, Masoud"https://www.zbmath.org/authors/?q=ai:ataei.masoud"Kazemi, Mahsa"https://www.zbmath.org/authors/?q=ai:kazemi.mahsa"Moreno Maza, Marc"https://www.zbmath.org/authors/?q=ai:moreno-maza.marcThe Extended Hensel Construction (EHC) is an algorithm designed by \textit{T. Sasaki} and \textit{F. Kako} [Japan J. Ind. Appl. Math. 16, No. 2, 257--285 (1999; Zbl 0941.12002)] for factoring univariate polynomials with power series coefficients. The goal of this algorithm was to provide an alternative approach to the classical NewtonPuiseux method for univariate power series coefficients. The EHC relies on Yun-Moses polynomials which make this algorithm inefficient in practice. In the paper under review, the authors show that the EHC requires only linear algebra and univariate polynomial arithmetic and report its complexity supported by favorable experimental results. Furthermore, they investigate two applications of this study. The first one is the computation of real branches of space curves and the second one deals with the computation of limits of real multivariate rational functions.
Reviewer: Amir Hashemi (Isfahan)Mirror symmetry of nonabelian Landau-Ginzburg orbifolds with loop type potentials.https://www.zbmath.org/1455.810412021-03-30T15:24:00+00:00"Mukai, Daichi"https://www.zbmath.org/authors/?q=ai:mukai.daichiSummary: We study the bigraded vector space structure of Landau-Ginzburg orbifolds with the permutation symmetries of variables. We calculate the formula of the Poincaré polynomial for a certain loop polynomial orbifoldized by a semidirect product of a diagonal symmetry group and cyclic permutation group and check that the generalization of Berglund-Hübsch transpose gives mirror symmetric pairs for these cases.A \(p\)-adically entire function with integral values on \(\mathbb{Q}_p\) and entire liftings of the \(p\)-divisible group \(\mathbb{Q}_p/\mathbb{Z}_p\). With an appendix by Maurizio Candilera.https://www.zbmath.org/1455.140482021-03-30T15:24:00+00:00"Baldassarri, Francesco"https://www.zbmath.org/authors/?q=ai:baldassarri.francescoSummary: We give a self-contained proof of the fact, discovered by \textit{F. Baldassarri} [``Interpretazione funzionale di certe iperalgebre e dei loro anelli di bivettori di Witt'', Tesi di laurea, Padova (1974) [2]] and proven by the author [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 321--331 (1975; Zbl 0308.12104)] with the methods of [\textit{I. Barsotti}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 1--25 (1964; Zbl 0121.16104)], that, for any prime number \(p\), there exists a power series \[\Psi= \Psi_p(T)\in T+ T^2\mathbb{Z}[[T]],\] which trivializes the addition law of the formal group of Witt covectors Barsotti [loc. cit.], [\textit{J.-M. Fontaine}, Astérisque 47--48, 262 p. (1977; Zbl 0377.14009), II.4], is \(p\)-adically entire and assumes values in \(\mathbb{Z}_p\) all over \(\mathbb{Q}_p\). We actually generalize, following a suggestion of M. Candilera, the previous facts to any fixed unramified extension \(\mathbb{Q}_q\) of \(\mathbb{Q}_p\) of degree \(f\), where \(q= p^f\) . We show that \(\Psi =\Psi_q\) provides a quasi-finite covering of the Berkovich affine line \(A^1_{\mathbb{Q}_p}\) by itself. We prove in Section 4 new strong estimates for the growth of \(\Psi\), in view of the application [\textit{F. Baldassarri}, ``The Artin-Hasse isomorphism of perfectoid open unit disks and a Fourier-type theory for continuous functions on \(\mathbb{Q}_p\)'', preprint [3]] to \(p\)-adic Fourier expansions on \(\mathbb{Q}_p\). We refer to Baldassarri [loc. cit. [3]] for the proof of a technical corollary (Proposition 4.13) which we apply here to locate the zeros of \(\Psi\) and to obtain its product expansion (Corollary 4.16). We reconcile the present discussion (for \(q= p\)) with the formal group proof given by Baldassarri [loc. cit. [2]], which takes place in the Fréchet algebra \(\mathbb{Q}_p\{x\}\) of the analytic additive group \(\mathbb{G}_{a,\mathbb{Q}_p}\) over \(\mathbb{Q}_p\). We show that, for any \(\lambda\in\mathbb{Q}^\times_p\), the closure \(\mathcal{E}^o_\lambda\) of \(\mathbb{Z}_p[\Psi(p^i x/\lambda)\mid i= 0,1,\dots]\) in \(\mathbb{Q}_p\{x\}\) is a Hopf algebra object in the category of Fréchet \(\mathbb{Z}_p\)-algebras. The special fiber of \(\mathcal{E}^o_\lambda\) is the affine algebra of the \(p\)-divisible group \(\mathbb{Q}_p/p\lambda \mathbb{Z}_p\) over \(\mathbb{F}_p\), while \(\mathcal{E}^o_\lambda[1/p]\) is dense in \(\mathbb{Q}_p\{x\}\). From \(\mathbb{Z}_p[\Psi(\lambda x)\mid\lambda\in \mathbb{Q}^\times_p]\) we construct a \(p\)-adic analog \(\mathcal{A}\mathcal{P}_{\mathbb{Q}_p}(\Sigma_\rho)\) of the algebra of Dirichlet series holomorphic in a strip \((\rho,\rho)\times i\mathbb{R}\subset\mathbb{C}\). We start developing this analogy. It turns out that the Banach algebra of almost periodic functions on \(\mathbb{Q}_p\) identifies with the topological ring of germs of holomorphic almost periodic functions on strips around \(\mathbb{Q}_p\).Comparison of stratified-algebraic and topological K-theory.https://www.zbmath.org/1455.141122021-03-30T15:24:00+00:00"Kucharz, Wojciech"https://www.zbmath.org/authors/?q=ai:kucharz.wojciech"Kurdyka, Krzysztof"https://www.zbmath.org/authors/?q=ai:kurdyka.krzysztofIn a previous paper [\textit{W. Kucharz} and \textit{K. Kurdyka}, J. Reine Angew. Math. 745, 105--154 (2018; Zbl 1410.14045)], the authors introduced and investigated stratified-algebraic vector bundles on real algebraic varieties. In the abstract of that paper they wrote: ``A stratification of \(X\) is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is \(X\). A topological vector bundle \(\xi\) on \(X\) is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification \(\mathcal{S}\) of \(X\) such that the restriction of \(\xi\) to each stratum \(S\) in \(\mathcal{S}\) is an algebraic vector bundle on \(S\). In particular, every algebraic vector bundle on \(X\) is stratified-algebraic.''
Stratified-algebraic vector bundles occupy an intermediate position between algebraic and topological vector bundles.
The present paper continues the line of research undertaken in the paper mentioned above and in [\textit{W. Kucharz}, J. Singul. 12, 92--104 (2015; Zbl 1307.14079)]. The authors give a characterization of the compact real algebraic varieties \(X\) having the following property: There exists a positive integer \(r\) such that for any constant rank topological vector bundle \(\xi\) on \(X\), the direct sum of \(r\) copies of \(\xi\) is isomorphic to a stratified-algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most \(8\) has this property. Their results are expressed in terms of \(K\)-theory.
In a broader context, the paper is also closely related to the investigations devoted to regulous functions on real algebraic sets (see, e.g. [\textit{G. Fichou} et al., J. Reine Angew. Math. 718, 103--151 (2016; Zbl 1390.14172)]).
Reviewer: Victor Zvonilov (Nizhny Novgorod)Frobenius powers.https://www.zbmath.org/1455.130102021-03-30T15:24:00+00:00"Hernández, Daniel J."https://www.zbmath.org/authors/?q=ai:hernandez.daniel-j"Teixeira, Pedro"https://www.zbmath.org/authors/?q=ai:teixeira.pedro"Witt, Emily E."https://www.zbmath.org/authors/?q=ai:witt.emily-eSummary: This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.Linear degenerations of flag varieties: partial flags, defining equations, and group actions.https://www.zbmath.org/1455.140952021-03-30T15:24:00+00:00"Cerulli Irelli, Giovanni"https://www.zbmath.org/authors/?q=ai:cerulli-irelli.giovanni"Fang, Xin"https://www.zbmath.org/authors/?q=ai:fang.xin"Feigin, Evgeny"https://www.zbmath.org/authors/?q=ai:feigin.evgeny"Fourier, Ghislain"https://www.zbmath.org/authors/?q=ai:fourier.ghislain"Reineke, Markus"https://www.zbmath.org/authors/?q=ai:reineke.markusSummary: We continue, generalize and expand our study of linear degenerations of flag varieties from [the authors, Math. Z. 287, No. 1--2, 615--654 (2017; Zbl 1388.14145)]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.Rings with trivial FML-invariant.https://www.zbmath.org/1455.141192021-03-30T15:24:00+00:00"Daigle, Daniel"https://www.zbmath.org/authors/?q=ai:daigle.danielSummary: Let \textbf{k} be a field of characteristic zero and \(B\) a commutative integral domain that is also a finitely generated \textbf{k}-algebra. It is well known that if \textbf{k} is algebraically closed and the ``field Makar-Limanov'' invariant \(\text{FML}(B)\) is equal to \textbf{k}, then \(B\) is unirational over \textbf{k}. This article shows that, when \textbf{k} is not assumed to be algebraically closed, the condition \(\text{FML}(B) = \textbf{k}\) implies that there exists a nonempty Zariski-open subset \(U\) of Spec \(B\) with the following property: for each prime ideal \(\mathfrak{p} \in U, the \kappa (\mathfrak{p})\)-algebra \(\kappa (\mathfrak{p}) \otimes_{\textbf{k}}B\) can be embedded in a polynomial ring in \(n\) variables over \(\kappa (\mathfrak{p})\), where \(n = \text{dim} B\) and \(\kappa (\mathfrak{p}) = B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}} \).Penalty function based critical point approach to compute real witness solution points of polynomial systems.https://www.zbmath.org/1455.650802021-03-30T15:24:00+00:00"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuan"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Reid, Greg"https://www.zbmath.org/authors/?q=ai:reid.greg-jSummary: We present a critical point method based on a penalty function for finding certain solution (witness) points on real solutions components of general real polynomial systems. Unlike other existing numerical methods, the new method does not require the input polynomial system to have pure dimension or satisfy certain regularity conditions.{
}This method has two stages. In the first stage it finds approximate solution points of the input system such that there is at least one real point on each connected solution component. In the second stage it refines the points by a homotopy continuation or traditional Newton iteration. The singularities of the original system are removed by embedding it in a higher dimensional space.{
}In this paper we also analyze the convergence rate and give an error analysis of the method. Experimental results are also given and shown to be in close agreement with the theory.
For the entire collection see [Zbl 1371.68008].On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups.https://www.zbmath.org/1455.140782021-03-30T15:24:00+00:00"Ebeling, Wolfgang"https://www.zbmath.org/authors/?q=ai:ebeling.wolfgang"Gusein-Zade, Sabir M."https://www.zbmath.org/authors/?q=ai:gusein-zade.sabir-mSummary: In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, \textit{P. Berglund} and \textit{M. Henningson} [Nucl. Phys., B 433, No. 2, 311--332 (1995; Zbl 0899.58068)] and \textit{P. Berglund} and \textit{T. Hübsch} [Nucl. Phys., B 393, No. 1--2, 377--391 (1993; Zbl 1245.14039)] considered a pair \((f,G)\) consisting of an invertible polynomial \(f\) and a finite abelian group \(G\) of its diagonal symmetries and associated to this pair a dual pair \((\tilde{f}, \tilde{G})\). A. Takahashi suggested a generalization of this construction to pairs \((f,G)\) where \(G\) is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper [Proc. Edinb. Math. Soc., II. Ser. 60, No. 1, 99--106 (2017; Zbl 1360.14107)], the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group \(G\): a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair \((f,G)\). We show that, for an abelian group \(G\), the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group \(G\) and the dual group \(\tilde{G}\) coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.A survey on maximal green sequences.https://www.zbmath.org/1455.130392021-03-30T15:24:00+00:00"Keller, Bernhard"https://www.zbmath.org/authors/?q=ai:keller.bernhard"Demonet, Laurent"https://www.zbmath.org/authors/?q=ai:demonet.laurentSummary: Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey recent progress on their existence and properties and give a representation-theoretic proof of Greg Muller's theorem stating that full subquivers inherit maximal green sequences. In the appendix, Laurent Demonet describes maximal chains of torsion classes in terms of bricks generalizing a theorem by Igusa.
For the entire collection see [Zbl 07314259].On the cohomological Hall algebra of the Kronecker quiver.https://www.zbmath.org/1455.141042021-03-30T15:24:00+00:00"Franzen, H."https://www.zbmath.org/authors/?q=ai:franzen.hans"Reineke, M."https://www.zbmath.org/authors/?q=ai:reineke.markusSummary: We give a short introduction to cohomological Hall algebras of quivers and describe the semistable cohomological Hall algebra of central slope of the Kronecker quiver in terms of generators and relations.
For the entire collection see [Zbl 07314259].Hodge theorem for the logarithmic de Rham complex via derived intersections.https://www.zbmath.org/1455.140142021-03-30T15:24:00+00:00"Hablicsek, Márton"https://www.zbmath.org/authors/?q=ai:hablicsek.martonSummary: In a beautiful paper, \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)] proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. \textit{K. Kato} [in: Algebraic analysis, geometry, and number theory: proceedings of the JAMI inaugural conference, held at Baltimore, MD, USA, May 16--19, 1988. Baltimore, MD: Johns Hopkins University Press. 191--224 (1989; Zbl 0776.14004)] generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections developed in [\textit{D. Arinkin} et al., Algebr. Geom. 4, No. 4, 394--423 (2017; Zbl 1401.14114)] and the author of this paper to give a new, geometric interpretation of the Hodge theorem for the logarithmic de Rham complex.The purity conjecture in type \(C\).https://www.zbmath.org/1455.050812021-03-30T15:24:00+00:00"Karpman, Rachel"https://www.zbmath.org/authors/?q=ai:karpman.rachelSummary: A collection \(\mathcal{C}\) of \(k\)-element subsets of \(\lbrace 1,2,\ldots ,m\rbrace\) is weakly separated if for each \(I, J \in \mathcal{C} \), when the integers \(1,2,\ldots ,m\) are arranged around a circle, there is a chord separating \(I\backslash J\) from \(J \backslash I\). \textit{S. Oh} et al. [Proc. Lond. Math. Soc. (3) 110, No. 3, 721--754 (2015; Zbl 1309.05182)] constructed a correspondence between weakly separated collections which are maximal by inclusion and reduced plabic graphs, a class of networks defined by \textit{A. Postnikov} [``Total positivity, Grassmannians and networks'', Preprint, \url{arXiv:math/0609764}] which give coordinate charts on the Grassmannian of \(k\)-planes in \(m\)-space. As a corollary, they proved Scott's Purity Conjecture, which states that a weakly separated collection is maximal by inclusion if and only if it is maximal by size. In this note, we describe maximal weakly separated collections corresponding to symmetric plabic graphs, which give coordinate charts on the Lagrangian Grassmannian, and prove a symmetric version of the Purity Conjecture.Derived categories and the genus of space curves.https://www.zbmath.org/1455.140392021-03-30T15:24:00+00:00"Macrì, Emanuele"https://www.zbmath.org/authors/?q=ai:macri.emanuele"Schmidt, Benjamin"https://www.zbmath.org/authors/?q=ai:schmidt.benjamin.1In this paper, using wall-crossing technieques of \textit{tilt stability} the authors give a generalization of the classical result of \textit{L. Gruson} and \textit{C. Peskine} [Lect. Notes Math. 687, 31--59 (1978; Zbl 0412.14011)] about the genus of curves in projective space, and indicate an approach to the Hartshorne-Hirschowitz conjecture.
The main result (Theorem 3.1) of the paper could be summarized as follows. Let \(X\) be a smooth projective threefold of Picard rank one such that any slope semistable sheaf in \(\mathrm{Coh}(X)\) satisfies the classical Bogomolov inequality, an Wd for any \(\nu_{\alpha, \beta}\)-semistable object \(E\) in the tilt heart \(\mathrm{Coh}^{\beta}(X)\) the inequality involving the third Chern character \(Q_{\alpha, \beta}(E) \geq 0\) holds. Let \(k \in \mathbb Z_{>0}\), \(d \in \frac 1 2 \mathbb Z_{>0}\) and \(C \subset X\) be an integral curve of degree \(d\). Assume \(H^0(X, I_C((k-1)H) = 0\) and \(d > k(k-1)\). Then \(\mathrm{ch}_3(I_C)\) has an upper bound \(H^3\cdot \left\{\frac{d^2}{2k} + \frac{dk}{2} - \epsilon(d,e) \right\}\). In characteristic zero, the theorem holds for \(X = \mathbb P^3\), principally polarized abelian threefolds of Picard rank one, and index two Fano threefolds of Picard rank one with degree one or two.
The basic idea of the proof is to consider the destabilizing sequence \(0 \rightarrow F \rightarrow E \rightarrow G \rightarrow 0\) for a \(\nu_{\alpha, \beta}\)-semistable object \(E\) with \(\mathrm{ch}_3(E)\) larger than expected. Thus \(\mathrm{ch}_1(F)\) has a lower and upper bound as \(\mathrm{ch}^{\beta}_1(F) > 0\) and \(\mathrm{ch}^{\beta}_1(G) > 0\). Also the Bogomolov inequalities of \(F\) and \(G\) lead to an upper and lower bound of \(\mathrm{ch}_2(F)\). Here we only have finitely many walls. Combined with bounds for \(\mathrm{ch}_3(F)\) and \(\mathrm{ch}_3(G)\) one can bound \(\mathrm{ch}_3(E)\) which would give a contradiction for all walls.
For the case of \(d \leq k(k-1)\) in projective space, we need to study walls above or below the specific wall \(W(I_C, \mathscr O(-f-4)[1])\) associated to the integral curve \(C\). In a smaller range around \(d\) with characteristic zero the authors shows \(\mathrm{ch}_3(I_C)\) has an upper bound depending on \(d\) and \(k\) if \(I_C\) is destabilized above or at the wall \(W\) in tilt stability. This result is one of sufficient conditions presented by the authors for the Hartshorne-Hirschowitz conjecture.
Reviewer: Hung-Yu Yeh (Taoyuan)The polymake interface in Singular and its applications.https://www.zbmath.org/1455.141162021-03-30T15:24:00+00:00"Epure, Raul"https://www.zbmath.org/authors/?q=ai:epure.raul"Ren, Yue"https://www.zbmath.org/authors/?q=ai:ren.yue"Schönemann, Hans"https://www.zbmath.org/authors/?q=ai:schonemann.hansSummary: Singular and polymake are computer algebra systems for research in algebraic geometry and polyhedral geometry respectively. We illustrate the implementation and the functionality of the polymake-interface in Singular and exhibit its application to the arithmetic of polyhedral divisors and the reconstruction of hypersurface singularities from the Milnor algebra.
For the entire collection see [Zbl 1371.68008].Logarithmic abelian varieties. VI: Local moduli and GAGF.https://www.zbmath.org/1455.140852021-03-30T15:24:00+00:00"Kajiwara, Takeshi"https://www.zbmath.org/authors/?q=ai:kajiwara.takeshi"Kato, Kazuya"https://www.zbmath.org/authors/?q=ai:kato.kazuya"Nakayama, Chikara"https://www.zbmath.org/authors/?q=ai:nakayama.chikaraSummary: This is Part VI of our series of papers on log abelian varieties. In this part, we study local moduli and GAGF of log abelian varieties.
For Part I--V see [the authors, Nagoya Math. J. 189, 63--138 (2008; Zbl 1169.14031); J. Math. Sci., Tokyo 15, No. 1, 69--193 (2008; Zbl 1156.14038); Nagoya Math. J. 210, 59--81 (2013; Zbl 1280.14008); Nagoya Math. J. 219, 9--63 (2015; Zbl 1329.14090); ibid. 64, 21--82 (2018; Zbl 1420.14102)].Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras.https://www.zbmath.org/1455.140432021-03-30T15:24:00+00:00"Neshitov, Alexander"https://www.zbmath.org/authors/?q=ai:neshitov.alexander"Petrov, Victor"https://www.zbmath.org/authors/?q=ai:petrov.viktor"Semenov, Nikita"https://www.zbmath.org/authors/?q=ai:semenov.nikita"Zainoulline, Kirill"https://www.zbmath.org/authors/?q=ai:zainoulline.kirillSummary: Let \(G\) be a split semisimple linear algebraic group over a field \(k_0\). Let \(E\) be a \(G\)-torsor over a field extension \(k\) of \(k_0\). Let \(\mathtt{h}\) be an algebraic oriented cohomology theory in the sense of \textit{M. Levine} and \textit{F. Morel} [Algebraic cobordism. Berlin: Springer (2007; Zbl 1188.14015)]. Consider a twisted form \(E / B\) of the variety of Borel subgroups \(G / B\) over \(k\).
Following the Kostant-Kumar results on equivariant cohomology of flag varieties
[\textit{B. Kostant} and \textit{S. Kumar}, J. Differ. Geom. 32, No. 2, 549--603 (1990; Zbl 0731.55005)] we establish an isomorphism between the Grothendieck groups of the \(\mathtt{h}\)-motivic subcategory generated by \(E / B\) and the category of finitely generated projective modules of certain Hecke-type algebra \(H\) which depends on the root datum of \(G\), on the torsor \(E\) and on the formal group law of the theory \(\mathtt{h}\).
In particular, taking \(\mathtt{h}\) to be the Chow groups with finite coefficients \(\mathbb{F}_p\) and \(E\) to be a generic \(G\)-torsor we prove that all finitely generated projective indecomposable submodules of an affine nil-Hecke algebra \(H\) of \(G\) with coefficients in \(\mathbb{F}_p\) are isomorphic to each other and correspond to the (non-graded) generalized Rost-Voevodsky motive for \((G, p)\).Full rank representation of real algebraic sets and applications.https://www.zbmath.org/1455.141152021-03-30T15:24:00+00:00"Chen, Changbo"https://www.zbmath.org/authors/?q=ai:chen.changbo"Wu, Wenyuan"https://www.zbmath.org/authors/?q=ai:wu.wenyuan"Feng, Yong"https://www.zbmath.org/authors/?q=ai:feng.yongSummary: We introduce the notion of the full rank representation of a real algebraic set, which represents it as the projection of a union of real algebraic manifolds \(V_{\mathbb {R}}(F_i)\) of \(\mathbb {R}^m\), \(m\geq n\), such that the rank of the Jacobian matrix of each \(F_i\) at any point of \(V_{\mathbb {R}}(F_i)\) is the same as the number of polynomials in \(F_i\).{
}By introducing an auxiliary variable, we show that a squarefree regular chain \(T\) can be transformed to a new regular chain \(C\) having various nice properties, such as the Jacobian matrix of \(C\) attains full rank at any point of \(V_{\mathbb {R}}(C)\). Based on a symbolic triangular decomposition approach and a numerical critical point technique, we present a hybrid algorithm to compute a full rank representation.{
}As an application, we show that such a representation allows to better visualize plane and space curves with singularities. Effectiveness of this approach is also demonstrated by computing witness points of polynomial systems having rank-deficient Jacobian matrices.
For the entire collection see [Zbl 1371.68008].K-semistability of optimal degenerations.https://www.zbmath.org/1455.320022021-03-30T15:24:00+00:00"Dervan, Ruadhaí"https://www.zbmath.org/authors/?q=ai:dervan.ruadhaiSummary: K-polystability of a polarized variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a `most destabilizing' degeneration. In this note we show that if such a degeneration exists, then the limiting scheme is itself relatively K-semistable.Trace ideals, normalization chains, and endomorphism rings.https://www.zbmath.org/1455.130202021-03-30T15:24:00+00:00"Faber, Eleonore"https://www.zbmath.org/authors/?q=ai:faber.eleonoreLet \(R\) be a reduced commutative Noetherian ring. Let \(M\) be a finitely generated reflexive \(R\)-module and \(R'\) a finite birational extension of \(R.\) It is shown that \(M\) is a \(R'\)-module if and only if \(\tau(M)\subset\mathcal{C}_{R'/R}\), where \(\tau(M)\) is the trace ideal of \(M\) and \(\mathcal{C}_{R'/R}\) is the conductor of \(R'\) in \(R.\) As a corollary it is shown that a reduced local complete ring of dimension \(1\) and embedding dimension \(2\) containing \(\mathbb{Q}\) is of finite CM-type if and only if there are finitely many possibilities for \(\tau(M),\) where \(M\) is a CM-moule over \(R\). Then one-dimensional local rings \((R,\mathfrak{m})\) such that their normalization is \(\mathrm{End}_R(\mathfrak{m})\) are studied. A criterion for this property in terms of the conductor ideal is given and it is shown that these rings are nearly Gorenstein.
Reviewer: Cristodor-Paul Ionescu (Bucureşti)Quantum singularity theory via cosection localization.https://www.zbmath.org/1455.141062021-03-30T15:24:00+00:00"Kiem, Young-Hoon"https://www.zbmath.org/authors/?q=ai:kiem.young-hoon"Li, Jun"https://www.zbmath.org/authors/?q=ai:li.jun.1Summary: We generalize the cosection localized Gysin map to intersection homology and Borel-Moore homology, which provides us with a purely topological construction of the Fan-Jarvis-Ruan-Witten invariants and some GLSM invariants.Dependent subsets of embedded projective varieties.https://www.zbmath.org/1455.141022021-03-30T15:24:00+00:00"Ballico, Edoardo"https://www.zbmath.org/authors/?q=ai:ballico.edoardoSummary: Let \(X\subset \mathbb{P}^r\) be an integral and non-degenerate variety. Set \(n:= \dim (X)\). Let \(\rho (X)''\) be the maximal integer such that every zero-dimensional scheme \(Z\subset X\) smoothable in \(X\) is linearly independent. We prove that \(X\) is linearly normal if \(\rho (X)''\ge \lceil (r+2)/2\rceil\) and that \(\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil \), unless either \(n=r\) or \(X\) is a rational normal curve.Correction to: ``Cohomology of the toric arrangement associated with \(A_n\)''.https://www.zbmath.org/1455.200262021-03-30T15:24:00+00:00"Bergvall, Olof"https://www.zbmath.org/authors/?q=ai:bergvall.olofCorrection to the author's paper [ibid. 21, No. 1, Paper No. 15, 14 p. (2019; Zbl 1454.20081)].Mukai's program (reconstructing a \(K3\) surface from a curve) via wall-crossing.https://www.zbmath.org/1455.140342021-03-30T15:24:00+00:00"Feyzbakhsh, Soheyla"https://www.zbmath.org/authors/?q=ai:feyzbakhsh.soheylaSummary: Let \(C\) be a curve of genus \(g=11\) or \(g\geq 13\) on a \(K3\) surface whose Picard group is generated by the curve class \([C]\). We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai's program to this situation: we show how to reconstruct the \(K3\) surface containing the curve \(C\) as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on \(C\).Parameterization of rational translational surfaces.https://www.zbmath.org/1455.141142021-03-30T15:24:00+00:00"Pérez-Díaz, Sonia"https://www.zbmath.org/authors/?q=ai:perez-diaz.sonia"Shen, Li-Yong"https://www.zbmath.org/authors/?q=ai:shen.liyongIn this paper, the authors deals with translation surfaces. Rational
translation surfaces are surfaces that can be rationally parametrized
as the addition of two curve parametrizations. Therefore, the
information on the surface lies on two rational curves contained in the surface.
The authors solve algorithmically the problem of
deciding whether an implicitely given rational surface is indeed a translation
surface and, in the affirmative case, they find two rational curves on
the surface such that its addition provides a parametrization of the surface.
Reviewer: Juan Rafael Sendra (Alcalá de Henares)Diophantine problems and \(p\)-adic period mappings.https://www.zbmath.org/1455.110932021-03-30T15:24:00+00:00"Lawrence, Brian"https://www.zbmath.org/authors/?q=ai:lawrence.brian"Venkatesh, Akshay"https://www.zbmath.org/authors/?q=ai:venkatesh.akshayIn the paper under review, the authors introduce a novel \(p\)-adic approach for establishing Zariski non-density results for varieties over number fields. They illustrate their methods by giving new proofs of the finiteness of solutions of the \(S\)-unit equation and of Falting's theorem (Mordell's conjecture). Moreover, they produce a result concerning the moduli space of degree \(d\) hypersurfaces in \(\mathbb P^n\), for sufficiently large \(d\) and \(n\). Note that the last result is well beyond the \(1\) dimensional case.
The method introduced in this paper has many similarities in spirit with Kim's non-abelian Chabatau [\textit{M. Kim}, Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006); Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl 1165.14020)]. Both methods use \(p\)-adic Hodge theory and \(p\)-adic period maps in an essential way, albeit in a different manner.
The rough idea is as follows: Let \(Y/K\) be a smooth variety over a number field and \(S\) a finite set of places of \(K\).
In place of Kim's non-abelian cohomology, the authors use a suitable smooth proper covering \(\pi:X\to Y\).
Assume that \(\pi\) extends to a smooth proper morphism of smooth \(\mathcal O_{K,S}\)-schemes, and choose a place \(v\notin S\), with residue characteristic \(p\), which satisfies some minor conditions.
Fix \(y_0\in Y(K_v)\). For each \(K_v\)-point of \(Y\) the de Rham cohomology of its fiber carries the structure of a filtered \(\phi\)-module over \(K_v\). This gives rise to a local period map with source the residue disc of \(Y(K_v)\) at \(y_0\).
The authors relate the local to the complex period map, by using the fact that in both situations the identification of the cohomology of nearby fibres can be given by the Gauss-Manin connection which can be defined over \(K\). Using this relation, the authors can control the fibers of the local period map under the assumption that the centraliser of the crystalline Frobenius is small in an appropriate sense which is related with the topological monodromy group which needs to be `big'.
Each \(S\)-integral point, corresponds to a \(K\)-point and the \(p\)-adic cohomology of the geometric fibre at that point gives a global Galois representation. Consider now the \(S\)-integral points in the residue disc at \(y_0\), whose associated global Galois representation is semisimple. The image under the local period map of these points, is contained in a finite union of orbits of the centraliser of the crystalline Frobenius on the `local period domain'. Once one has enough control on the fibers of the local period map and the image of the \(S\)-integral points, the method gives Zariski non-density results.
For the method to go through, one needs to exhibit a family, such that (a) one can show that the centraliser of the crystalline Frobenius is small in an appropriate sense and (b) one can control the \(S\)-integral points for which the associated global representation in not semisimple (conjecturally there are no such points).
The authors successfully implement their methods for the three results mentioned in the first paragraph by using a variant of the Legendre family for the \(S\)-unit equation, by constructing a suitable family they call Kodaira-Parshin family for the Mordell conjecture, whereas for the moduli space they use the universal family. The methods introduced in this paper have many more potential applications. On the effectivity side, an interesting question is whether and to what extent these methods can become explicit and algorithmic.
Reviewer: Evis Ieronymou (Nicosia)Nil-Hecke algebras and Whittaker \(\mathfrak{D}\)-modules.https://www.zbmath.org/1455.200302021-03-30T15:24:00+00:00"Ginzburg, Victor"https://www.zbmath.org/authors/?q=ai:ginzburg.viktor-l|ginzburg.victorSummary: Given a semisimple group \(G\), Kostant and Kumar defined a nil-Hecke algebra that may be viewed as a degenerate version of the double affine nil-Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical subalgebra of the nil-Hecke algebra with a Whittaker type quantum Hamiltonian reduction of the algebra of differential operators on \(G\). This result has an interpretation in terms of geometric Satake and the Langlands dual group. Specifically, the isomorphism provides a bridge between very differently looking descriptions of equivariant Borel-Moore homology of the affine flag variety (due to Kostant and Kumar) and of the affine Grassmannian (due to Bezrukavnikov and Finkelberg), respectively.
It follows from our result that the category of Whittaker \(\mathfrak{D}\)-modules on \(G\), considered by Drinfeld, is equivalent to the category of holonomic modules over the nil-Hecke algebra, and it is also equivalent to a certain subcategory of the category of Weyl group equivariant holonomic \(\mathfrak{D}\)-modules on the maximal torus.
For the entire collection see [Zbl 1412.22001].Variations on a theorem of Tate.https://www.zbmath.org/1455.110032021-03-30T15:24:00+00:00"Patrikis, Stefan"https://www.zbmath.org/authors/?q=ai:patrikis.stefan-tLet \(F\) be a number field and \(\Gamma_F =\mathrm{Gal}(\overline{F}/F)\) its absolute Galois group. Tate's theorem that
\(H^2(\Gamma_f, {\mathbb Q}/{\mathbb Z})\) vanishes implies the fundamental fact that continuous projective representations \(\Gamma_F \mapsto\mathrm{PGL}_n({\mathbb C})\) lift to \(\mathrm{GL}_n({\mathbb C})\). The present book studies the corresponding problem when \(\Gamma_F\) is replaced by groups such as the automorphic Langlands group \({\mathcal L}_F\) or the motivic Galois group \({\mathcal G}_F\).
Since these groups are ``hypothetical'' objects only believed to exist, the first task is translating the similarly hypothetical lifting property into concrete and well-defined questions about automorphic representations. This is where fundamental conjectures due to Fontaine-Mazur, Langlands, Serre and Grothendieck play a role; informally, we expect bijections between suitable automorphic representations of \(\mathrm{GL}_n({\mathbb A}_F)\), irreducible Galois representations \(\Gamma_F \mapsto \mathrm{GL}_n(\overline{{\mathbb Q}}_\ell)\) and irreducible pure motives over \(F\) with \(\overline{\mathbb Q}\)-coefficients, all characterized by certain local compatibility conditions.
The first chapter is devoted to a discussion of the relationship between all these conjectures. Chapter 2 contains preliminary results and examples arising from the special cases of \(\mathrm{GL}_1\) and \(\mathrm{GL}_2\); Chapter 3 presents the general automorphic and Galois-theoretic lifting theorems and results about monodromy groups of Galois representations. Chapter 4 discusses motivic lifting problems and the generalized Kuga-Satake conjecture.
Reviewer: Franz Lemmermeyer (Jagstzell)Donaldson-Thomas invariants from tropical disks.https://www.zbmath.org/1455.141032021-03-30T15:24:00+00:00"Cheung, Man-Wai"https://www.zbmath.org/authors/?q=ai:cheung.man-wai"Mandel, Travis"https://www.zbmath.org/authors/?q=ai:mandel.travisSummary: We prove that the quantum DT-invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in terms of stability conditions, plus a new version of the description of scattering diagrams in terms of tropical disk counts. The weights with which the tropical disks are counted are expressed in terms of motivic integrals of certain quiver flag varieties. We also show via explicit counterexample that Hall algebra broken lines do not result in consistent Hall algebra theta functions, i.e., they violate the extension of a lemma of Carl-Pumperla-Siebert from the classical setting.Preface to the special issue.https://www.zbmath.org/1455.140022021-03-30T15:24:00+00:00"Kashiwara, Masaki (ed.)"https://www.zbmath.org/authors/?q=ai:kashiwara.masaki"Tamagawa, Akio (ed.)"https://www.zbmath.org/authors/?q=ai:tamagawa.akio"Arakawa, Tomoyuki (ed.)"https://www.zbmath.org/authors/?q=ai:arakawa.tomoyuki"Hasegawa, Masahito (ed.)"https://www.zbmath.org/authors/?q=ai:hasegawa.masahito"Kumagai, Takashi (ed.)"https://www.zbmath.org/authors/?q=ai:kumagai.takashi"Makino, Kazuhisa (ed.)"https://www.zbmath.org/authors/?q=ai:makino.kazuhisa"Mochizuki, Takuro (ed.)"https://www.zbmath.org/authors/?q=ai:mochizuki.takuro"Mukai, Shigeru (ed.)"https://www.zbmath.org/authors/?q=ai:mukai.shigeru"Nakajima, Hiraku (ed.)"https://www.zbmath.org/authors/?q=ai:nakajima.hiraku"Nakanishi,Kenji (ed.)"https://www.zbmath.org/authors/?q=ai:nakanishi.kenji.2|nakanishi.kenji.1"Tomotada, Ohtsuki (ed.)"https://www.zbmath.org/authors/?q=ai:tomotada.ohtsuki"Ono, Kaoru (ed.)"https://www.zbmath.org/authors/?q=ai:ono.kaoru"Ozawa, Narutaka (ed.)"https://www.zbmath.org/authors/?q=ai:ozawa.narutaka"Yamada, Michio (ed.)"https://www.zbmath.org/authors/?q=ai:yamada.michioFrom the text: There are two main reasons for publishing this series of papers of S. Mochizuki on Inter-universal Teichmüller theory in a special
issue. One is their volume and importance. The other is to avoid the conflict of
interest that arises because the author is Editor-in-Chief of PRIMS.
As a general rule, when a paper is submitted to PRIMS by a member of
the Editorial Board, the member should be entirely excluded from the editorial
committee charged with handling it. When Mochizuki became Editor-in-Chief of
PRIMS in April 2012, the Editorial Board further decided that, in the case of his
submission, they would form a special committee to handle it, excluding him and
with an Editor-in-Chief substituting for him. When he submitted the present series
of papers on August 30, 2012, Akio Tamagawa took the job of Editor-in-Chief of
the special committee. Masaki Kashiwara later joined the committee, and he and
Tamagawa served as co-Editors-in-Chief.
Several mathematicians kindly accepted an invitation to referee the papers;
we are extremely grateful to them for their efforts and patience. Based on their
reports, we had numerous editorial meetings. In particular because of the total
length of the series of papers, it took a long time for the Editorial Committee to
arrive at the final decision of acceptance.On the Medvedev-Scanlon conjecture for minimal threefolds of nonnegative Kodaira dimension.https://www.zbmath.org/1455.140272021-03-30T15:24:00+00:00"Bell, Jason P."https://www.zbmath.org/authors/?q=ai:bell.jason-p"Ghioca, Dragos"https://www.zbmath.org/authors/?q=ai:ghioca.dragos"Reichstein, Zinovy"https://www.zbmath.org/authors/?q=ai:reichstein.zinovy-b"Satriano, Matthew"https://www.zbmath.org/authors/?q=ai:satriano.matthewSummary: Motivated by work of Zhang from the early `90s, \textit{A. Medvedev} and \textit{T. Scanlon} [Ann. Math. (2) 179, No. 1, 81--177 (2014; Zbl 1347.37145)] formulated the following conjecture. Let \(F\) be an algebraically closed field of characteristic 0 and let \(X\) be a quasiprojective variety defined over \(F\) endowed with a dominant rational self-map \(\phi \).
Then there exists a point \(x \in X(F)\) with Zariski dense orbit under \(\phi\) if and only if \(\phi\) preserves no nontrivial rational fibration, i.e., there exists no nonconstant rational functions \(f\in F(X)\) such that \(\phi^\ast(f)=f\).
The Medvedev-Scanlon conjecture holds when \(F\) is uncountable. The case where \(F\) is countable (e.g., \(F = \overline{\mathbb Q})\) is much more difficult; here the Medvedev-Scanlon conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension 0. Our results are most definitive in dimension 3.Moduli of formal torsors.https://www.zbmath.org/1455.140242021-03-30T15:24:00+00:00"Tonini, Fabio"https://www.zbmath.org/authors/?q=ai:tonini.fabio"Yasuda, Takehiko"https://www.zbmath.org/authors/?q=ai:yasuda.takehikoSummary: We construct the moduli stack of torsors over the formal punctured disk in characteristic \(p > 0\) for a finite group isomorphic to the semidirect product of a \(p\)-group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.Gorenstein modifications and \(\mathbb{Q}\)-Gorenstein rings.https://www.zbmath.org/1455.140052021-03-30T15:24:00+00:00"Dao, Hailong"https://www.zbmath.org/authors/?q=ai:dao.hailong"Iyama, Osamu"https://www.zbmath.org/authors/?q=ai:iyama.osamu"Takahashi, Ryo"https://www.zbmath.org/authors/?q=ai:takahashi.ryo"Wemyss, Michael"https://www.zbmath.org/authors/?q=ai:wemyss.michaelSummary: Let \(R\) be a Cohen-Macaulay normal domain with a canonical module \(\omega_R\). It is proved that if \(R\) admits a noncommutative crepant resolution (NCCR), then necessarily it is \(\mathbb{Q}\)-Gorenstein. Writing \(S\) for a Zariski local canonical cover of \(R\), a tight relationship between the existence of noncommutative (crepant) resolutions on \(R\) and \(S\) is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-terminal, and the Auslander-Esnault classification of two-dimensional CM-finite algebras can be deduced from [\textit{R. O. Buchweitz} et al., Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)].Explicit resolution of weak wild quotient singularities on arithmetic surfaces.https://www.zbmath.org/1455.140722021-03-30T15:24:00+00:00"Obus, Andrew"https://www.zbmath.org/authors/?q=ai:obus.andrew"Wewers, Stefan"https://www.zbmath.org/authors/?q=ai:wewers.stefanSummary: A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic \(p\) fiber is a \(p\)-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of \textit{D. Lorenzini} [Algebra Number Theory 8, No. 2, 331--367 (2014; Zbl 1332.14029)]. Along the way, we give a valuation-theoretic criterion for a normal snc-model of \(\mathbb{P}^1\) over a discretely valued field to be regular.On rigid varieties with projective reduction.https://www.zbmath.org/1455.140502021-03-30T15:24:00+00:00"Li, Shizhang"https://www.zbmath.org/authors/?q=ai:li.shizhangSummary: In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The Main Theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that \(p\)-adic Hopf varieties will never have a projective reduction. The proof of our Main Theorem uses the theory of moduli of semistable coherent sheaves.A universal coefficient theorem with applications to torsion in Chow groups of Severi-Brauer varieties.https://www.zbmath.org/1455.140152021-03-30T15:24:00+00:00"Mackall, Eoin"https://www.zbmath.org/authors/?q=ai:mackall.eoinSummary: For any variety \(X\), and for any coefficient ring \(S\), we define the \(S\)-topological filtration on the Grothendieck group of coherent sheaves \(G(X) \otimes S\) with coefficients in \(S\). The \(S\)-topological filtration is related to the topological filtration by means of a universal coefficient theorem. We apply this observation in the case \(X\) is a Severi-Brauer variety to obtain new examples of torsion in the Chow groups of \(X\).Handbook for mirror symmetry of Calabi-Yau and Fano manifolds. Selected papers based on the presentations at the conference, Taipei, Taiwan, January 06--10, 2014.https://www.zbmath.org/1455.140012021-03-30T15:24:00+00:00"Ji, Lizhen (ed.)"https://www.zbmath.org/authors/?q=ai:ji.lizhen"Wu, Baosen (ed.)"https://www.zbmath.org/authors/?q=ai:wu.baosen"Yau, Shing-Tung (ed.)"https://www.zbmath.org/authors/?q=ai:yau.shing-tungPublisher's description: In algebraic geometry and theoretical physics, mirror symmetry refers to the relationship between two Calabi-Yau manifolds which appear very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
Mathematicians became interested in mirror symmetry around 1990, when it was shown that mirror symmetry could be used to count rational curves on a Calabi-Yau manifold, thus solving a long-standing problem.
Today, mirror symmetry is a fundamental tool for doing calculations in string theory, and it has been used to understand aspects of quantum field theory, the formalism that physicists use to describe elementary particles. Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.
This handbook surveys recent developments in mirror symmetry. It presents papers based on selected lectures given at a 2014 Taipei conference on ``Calabi-Yau geometry and mirror symmetry,'' along with other contributions from invited authors.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Chan, Kwokwai}, SYZ mirror symmetry for toric varieties, 1-32 [Zbl 1440.14186]
\textit{Chen, Xi; Lewis, James D.}, Differential equations associated to normal functions, and the transcendental regulator for a \(K3\) surface and its self-product, 33-57 [Zbl 1440.14178]
\textit{Golyshev, V. V.}, Techniques to compute monodromy of differential equations of mirror symmetry, 59-87 [Zbl 1440.14188]
\textit{Hosono, Shinobu; Takagi, Hiromichi}, Mirror symmetry and projective geometry of Fourier-Mukai partners, 89-130 [Zbl 1440.14189]
\textit{Iritani, Hiroshi}, Quantum D-modules of toric varieties and oscillatory integrals, 131-147 [Zbl 1440.14190]
\textit{Kanazawa, Atsushi}, Degenerations and Lagrangian fibrations of Calabi-Yau manifolds, 149-204 [Zbl 1440.14185]
\textit{Lau, Siu-Cheong}, Generalized SYZ and homological mirror symmetry, 205-227 [Zbl 1440.14191]
\textit{Manin, Yuri I.}, Mirrors, functoriality, and derived geometry, 229-251 [Zbl 1440.14012]
\textit{Rubin-Blaier, Netanel}, Lectures on operads, homology operations, and mirror symmetry, 253-463 [Zbl 1440.18001]
\textit{Toda, Yukinobu}, S-duality conjecture for Calabi-Yau 3-folds, 465-482 [Zbl 1440.14095]
\textit{Ueda, Kazushi}, Mirror symmetry and \(K3\) surfaces, 483-521 [Zbl 1440.14195]
\textit{Zinger, Aleksey}, Some conjectures on the asymptotic behavior of Gromov-Witten invariants, 523-550 [Zbl 1440.14262]Projective bundle formula for Heller's relative \(K_0\).https://www.zbmath.org/1455.140172021-03-30T15:24:00+00:00"Sadhu, Vivek"https://www.zbmath.org/authors/?q=ai:sadhu.vivekSummary: In this article, we study the Heller relative \(K_0\) group of the map \(\mathbb{P}_X^r \to \mathbb{P}_S^r \), where \(X\) and \(S\) are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative \(K_0\), provided \(X\) is flat over \(S\). As a corollary, we get a description of the relative group \(K_0 (\mathbb{P}_X^r \to \mathbb{P}_S^r)\) in terms of generators and relations, provided \(X\) is affine and flat over \(S\).Generic injectivity of the Prym map for double ramified coverings.https://www.zbmath.org/1455.140642021-03-30T15:24:00+00:00"Naranjo, Juan Carlos"https://www.zbmath.org/authors/?q=ai:naranjo.juan-carlos"Ortega, Angela"https://www.zbmath.org/authors/?q=ai:ortega.angelaSummary: We consider the Prym map for double coverings of curves of genus \( g\) ramified at \( r>0\) points; that is, the map associated with a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less than or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with (a) \( g=2\), \( r=6\), and (b) \( g= 5\), \( r=2\). In the first case the proof is constructive and can be extended to the range \( r\geq \max \{6,\frac 23(g+2) \}\). For (b) we study the fiber along the locus of the intermediate Jacobians of cubic threefolds to conclude the generic injectivity. This completes the work of Marcucci and Pirola, who proved this theorem for all the other cases except for the bielliptic case \( g=1\) (solved later by Marcucci and the first author); the case \( g=3, r=4\) considered previously by \textit{D. S. Nagaraj} and \textit{S. Ramanan} [Duke Math. J. 80, No. 1, 157--194 (1995; Zbl 0879.14020)]; and by \textit{F. Bardelli} et al. [Compos. Math. 96, No. 2, 115--147 (1995; Zbl 0864.14027)], where the degree of the map is 3. We close with an appendix by Alessandro Verra with an independent result, the rationality of the moduli space of coverings with \( g=2,r=6\), whose proof is self-contained.Affine flag varieties and quantum symmetric pairs. II: Multiplication formula.https://www.zbmath.org/1455.170152021-03-30T15:24:00+00:00"Fan, Zhaobing"https://www.zbmath.org/authors/?q=ai:fan.zhaobing"Li, Yiqiang"https://www.zbmath.org/authors/?q=ai:li.yiqiang\textit{A. A. Beilinson} et al. [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)] that partial flag varieties of type \(A\) are a geometric setting for quantum \(\mathfrak{gl}_n\). ``For other classical partial flag varieties, it is only known recently that they are governed by coideal subalgebras of quantum \(\mathfrak{gl}_n\)''.
In [\textit{Z. Fan} et al., Affine flag varieties and quantum symmetric pairs.Mem. Am. Math. Soc. 1285, 123 p. (2020; Zbl 1452.17001)] the authors provided a description of the convolution algebra of affine partial flag varieties of type \(C\). In this paper it is proved that it is controlled by certain coideal subalgebra of quantum affine \(\mathfrak{gl}_n\) and gives a canonical bases for the latter algebras parametrized by certain tridiagonal matrices.
There is yet another natural set of multiplicative generators consisting of the tridiagonal standard basis elements, in the present paper a multiplication formula for this base is obtained.
Reviewer: Dmitry Artamonov (Moskva)Fano threefolds as equivariant compactifications of the vector group.https://www.zbmath.org/1455.140712021-03-30T15:24:00+00:00"Huang, Zhizhong"https://www.zbmath.org/authors/?q=ai:huang.zhizhong"Montero, Pedro"https://www.zbmath.org/authors/?q=ai:montero.pedro-jSummary: In this article, we determine all equivariant compactifications of the three-dimensional vector group \(\mathbf{G}_a^3\) that are smooth Fano threefolds with Picard number greater than or equal to two.On sofic groups, Kaplansky's conjectures, and endomorphisms of pro-algebraic groups.https://www.zbmath.org/1455.370132021-03-30T15:24:00+00:00"Phung, Xuan Kien"https://www.zbmath.org/authors/?q=ai:phung.xuan-kienSummary: Let \(G\) be a group. Let \(X\) be a connected algebraic group over an algebraically closed field \(K\). Denote by \(A = X(K)\) the set of \(K\)-points of \(X\). We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over \((G, X, K)\). They are cellular automata \(\tau : A^G \to A^G\) whose local defining map is induced by a homomorphism of algebraic groups \(X^M \to X\) where \(M \subset G\) is a finite memory set of \(\tau \). Our first result is that when \(G\) is sofic, such an algebraic group cellular automaton \(\tau\) is invertible whenever it is injective and \(\text{char}(K) = 0\). When \(G\) is amenable, we show that an algebraic group cellular automaton \(\tau\) is surjective if and only if it satisfies a weak form of pre-injectivity called \((\bullet)\)-pre-injectivity. This yields an analogue of the classical Moore-Myhill Garden of Eden theorem. We also introduce the near ring \(R(K, G)\) which is \(K [ X_g : g \in G]\) as an additive group but the multiplication is induced by the group law of \(G\). The near ring \(R(K, G)\) contains naturally the group ring \(K [G]\) and we extend Kaplansky's conjectures to this new setting. Among other results, we prove that when \(G\) is an orderable group, then all one-sided invertible elements of \(R(K, G)\) are trivial, i.e., of the form \(a X_g + b\) for some \(g \in G, a \in K^\ast \), and \(b \in K\). This in turns allows us to show that when \(G\) is locally residually finite and orderable (e.g. \( \mathbb{Z}^d\) or a free group), all injective algebraic cellular automata \(\tau : \mathbb{C}^G \to \mathbb{C}^G\) are of the form \(\tau(x)(h) = a x( g^{- 1} h) + b\) for all \(x \in \mathbb{C}^G, h \in G\) for some \(g \in G, a \in \mathbb{C}^\ast \), and \(b \in \mathbb{C} \).Smooth affine model for the framed correspondences spectrum.https://www.zbmath.org/1455.140412021-03-30T15:24:00+00:00"Druzhinin, A."https://www.zbmath.org/authors/?q=ai:druzhinin.andrei-eSummary: Morel-Voevodsky's unstable pointed motivic homotopy category \(\mathbf{H}_{\bullet}(k)\) over an infinite perfect field is considered. For a smooth affine scheme \(Y\) over \(k\), a smooth ind-scheme \(F_l(Y)\) and an open subscheme \(E_l(Y)\) are constructed for all \(l > 0\), so that the motivic space \(F_l(Y)/E_l(Y)\) is equivalent in \(\mathbf{H}_{\bullet}(k)\) to the motivic space \({\Omega}_{{\mathbb{P}}^1}^{\infty}\sum \limits_{\mathbb{P}^1}^{\infty}\left(Y\times{T}^l\right)\), \(T=\left(\mathbb{A}^1/\mathbb{A}^1-0\right)\), \(l>0 \). The construction is not functorial on the category of affine schemes but is functorial on the category of so-called framed schemes constructed for this purpose.Spectral invariants with bulk, quasi-morphisms and Lagrangian Floer theory.https://www.zbmath.org/1455.530012021-03-30T15:24:00+00:00"Fukaya, Kenji"https://www.zbmath.org/authors/?q=ai:fukaya.kenji"Oh, Yong-Geun"https://www.zbmath.org/authors/?q=ai:oh.yong-geun"Ohta, Hiroshi"https://www.zbmath.org/authors/?q=ai:ohta.hiroshi"Ono, Kaoru"https://www.zbmath.org/authors/?q=ai:ono.kaoruThis monograph constructs a very general new machine for building spectral invariants associated to (general) closed symplectic manifolds. As a consequence, the authors are able to prove powerful results on the groups of Hamiltonian diffeomophisms of many symplectic manifolds.
We recall what is meant by spectral invariants. Associated to a closed symplectic manifold \((M,\omega)\) and a Hamiltonian diffeomorphism \(H\), there is the Hamiltonian Floer homology group \(HF(M,H)\) obtained by counting solutions of the Cauchy-Riemann equation with a perturbation from the Hamiltonian. Said differently, \(HF(M,H)\) is obtained by counting formal trajectories of the action functional and, as such, admits a filtration by the action functional. Simultaneously, there is a natural isomorphism (where coefficients are taken in a Novikov ring \(\Lambda\)):
\[
\mathcal{P}_{H} \colon H_*(M;\Lambda) \to HF_*(M,H),
\]
of \textit{S. Piunikhin} [Quantum and Floer cohomology have the same ring structure. Ann Arbor, MI: MIT (PhD Thesis) (1996)]. This enables one to consider the homology of \(M\) itself as interacting with the action filtration. It turns out to be more natural to work with the quantum cohomology, using Poincaré duality:
\[
QH^*(M;\Lambda)\cong H_*(M;\Lambda),
\]
to then define a filtration of \(QH^*(M)\); for a class \(a\in QH^*(M)\), \textit{Y.-G. Oh} [J. Korean Math. Soc. 42, No. 1, 65--83 (2005; Zbl 1075.53088)] defined \(\rho(H;a)\) as the minimal filtration level in which \(a\) can be represented. These numbers \(\rho(H;a)\) are called the spectral invariants of \(H\); it turns out that they have a remarkable collection of good properties; among other things they are compatible with the Hofer norm and are \(C^0\)-continuous (in \(H\)). Oh [loc. cit.] also showed that for a path of normalized Hamiltonians (determining a path in the group of Hamiltonian diffeomorphisms \(\mathrm{Ham}(M,\omega)\)), the invariant \(\rho(H;a)\) only depends on the homotopy class of the involved path, and so descends to an invariant on the universal cover \(\widetilde{\mathrm{Ham}}(M;\omega)\), still called \(\rho(\tilde{\psi};a)\) for \(\tilde{\psi}\in \widetilde{\mathrm{Ham}}(M,\omega)\).
Entov-Polterovich showed that the spectral invariants, in turn, defined quasi-morphisms on \(\widetilde{\mathrm{Ham}}(M;\omega)\), under the following circumstances. Recall that a quasi-morphism on a group \(G\) is a function \(r: G\to \mathbb{R}\) which is a homomorphism up to a bounded error term, i.e., \(r\) for which there exists \(R>0\) such that:
\[
|r(fg)-r(f)-r(g)|\leq R
\]
for all \(f,g\in G\). Indeed, let \(e\) be a unit of a direct factor of \(QH^*(M;\Lambda)\); they proved that:
\[
\mu_e(\tilde{\psi}):=-\mathrm{vol}_{\omega}(M)\lim_{n\to \infty}\frac{\rho(\tilde{\psi}^n;e)}{n}
\]
is both well-defined and a homogeneous quasi-morphism.
The existence of a single quasi-morphism is already a very powerful statement about \(\widetilde{\mathrm{Ham}}(M;\omega)\); indeed, lower bounds for the Hofer norm of \(\tilde{\psi}\in\widetilde{\mathrm{Ham}}(M,\omega)\) naturally follow from the existence of a quasi-morphism, among many other consequences.
Indeed, the work of Ohta and Entov-Polterovich (among many other further developments of this work -- note that the early chapters of the monograph under review provide a nice development of the history) has already had a large impact on the study of dynamics on symplectic manifolds. However, in previous works (though cf. [\textit{M. Usher}, Isr. J. Math. 184, 1--57 (2011; Zbl 1253.53085)]), only a restrictive class of symplectic manifolds has been allowed in the place of \(M\).
The book generalizes ideas that have been executed in the case of (closed) semi-positive symplectic manifolds (among other contexts) to the case of general closed symplectic manifolds. In order to work at this level of generality, the authors use the language of Kuranishi atlases and the virtual fundamental cycle technique, as they have developed in their books on Lagrangian intersection Floer theory. However, the authors go to great length to make the usage digestible. Indeed, it is possible to read the book without being an expert on those works, and applications of this technical machinery (which are frequent) are carefully explained in the present work. As such, the work is an excellent instance of the application of the ideas from their earlier technical work.
However, in addition to bringing virtual fundamental cycle technology into the study of spectral invariants, the authors pursue two main new ideas. First, instead of working directly with the quantum cohomology above, they consider bulk deformations of the quantum cohomology by ambient cycles of symplectic manifolds. The second, and most fundamental, new addition is the use of the open-closed Gromov-Witten-Floer map, to go between working with the quantum cohomology ring and concrete Lagrangian Floer cohomology problems. These added techniques enable the authors to produce far more examples than could previously be considered. In particular, they are able to show that for many symplectic manifolds, there exists a continuum of independent quasi-morphisms on \(\widetilde{\mathrm{Ham}}(M,\omega)\). Subsequent to the appearance of this work on the arxiv, the work in this monograph has since been built on by many other authors to extend the reach of examples.
In outline, the work is set up as follows, in several parts. In the first part, the authors review the construction of Hamiltonian Floer homology and the spectral invariants of Oh. The second part contains a review of the quantum homology with bulk deformation. Already at this early point in the book, the authors are able to give the definition of the spectral invariants with bulk, the main object of interest. The construction here largely follows the same outline as Oh's outline, but is slightly more technical. The key formal properties of the spectral invariants with bulk are also established in this section, in particular, the triangle inequality follows from a pair-of-pants product argument.
In the third part, the authors define (spectral) partial symplectic quasi-states and spectral quasi-morphisms. We will not recall the technical definition here, but suffice it to say that these notions are a rather stricter notion of `measuring nondisplaceability' of subsets. They then show that the recipe of Entov-Polterovich can be extended to extract spectral quasi-morphisms and spectral partial symplectic quasi-states. Parts 4--5 address the applications to heavy and superheavy Lagrangian submanifolds. The book is closed out by an extensive appendix, giving technical details of some of the underlying constructions.
In spite of the technical complexity of the work, this monograph has a very clear presentation and provides spectacular applications of the techniques the authors have developed over a series of papers. This book is suitable for researchers and graduate students in sympletic topology, and it will be of great interest to all readers of FOOO's earlier books.
Reviewer: Matthew Stoffregen (Cambridge)Reduction of divisors for classical superintegrable \(\mathrm{GL}(3)\) magnetic chain.https://www.zbmath.org/1455.140702021-03-30T15:24:00+00:00"Tsiganov, A. V."https://www.zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovichSummary: Separated variables for a classical \(\mathrm{GL}(3)\) magnetic chain are coordinates of a generic positive divisor \(D\) of degree \(n\) on a genus \(g\) non-hyperelliptic algebraic curve. Because \(n > g\), this divisor \(D\) has unique representative \(\rho(D)\) in the Jacobian, which can be constructed by using \(\dim|D| = n - g\) steps of Abel's algorithm. We study the properties of the corresponding chain of divisors and prove that the classical \(\mathrm{GL}(3)\) magnetic chain is a superintegrable system with \(\dim|D| = 2\) superintegrable Hamiltonians.
{\copyright 2020 American Institute of Physics}Gröbner bases and the Cohen-Macaulay property of Li's double determinantal varieties.https://www.zbmath.org/1455.130222021-03-30T15:24:00+00:00"Fieldsteel, Nathan"https://www.zbmath.org/authors/?q=ai:fieldsteel.nathan"Klein, Patricia"https://www.zbmath.org/authors/?q=ai:klein.patriciaLi has defined \textit{double determinantal varieties} as follows (see also
[\textit{H. Nakajima}, in: Proceedings of the 49th symposium on ring theory and representation theory, Osaka Prefecture University, Osaka, Japan, August 31 -- September 3, 2016. Shimane: Symposium on Ring Theory and Representation Theory Organizing Committee. 96--114 (2017; Zbl 1437.17007); \textit{V. Ginzburg}, Sémin. Congr. 24, 145--219 (2012; Zbl 1305.16009)]): They are determined by the vanishing of the minors of some size \(s\) in a concatenation of finitely many \(m \times n\) matrices of indeterminates glued along their size \(m\) edges, together with the vanishing of minors of a possibly different size \(t\) in a concatenation of the same matrices along their length \(n\) edges. These are related to classical determinantal varieties, and more generally to ladder determinantal varieties, and even more generally to mixed determinantal varieties. All of these situations have been carefully studied, with results showing Gröbner bases, irreducibility, Cohen-Macaulayness, normality and glicciness. The list of authors who have studied all of these settings is too big to repeat here, but we mention in particular the work of
[\textit{E. Gorla}, J. Pure Appl. Algebra 211, No. 2, 433--444 (2007; Zbl 1128.14035)], whose methods are used in part in this paper. Liaison methods play an important role in the proofs, and in particular a key lemma from a paper of
\textit{E. Gorla} et al. [J. Algebra 384, 110--134 (2013; Zbl 1315.13042)].
The main result in this paper is the following: The natural generators of a double determinantal ideal form a Gröbner basis under any diagonal term order. Double determinantal varieties are irreducible, normal, and arithmetically Cohen-Macaulay. In addition, they are glicci. This proves a conjecture of Li. The authors point out that a different, more elementary proof of Li's conjecture by
\textit{L. Li} and \textit{I. L. J. Illian} [``Nakajima's quiver varieties and triangular bases of cluster algebras'', Preprint] precedes this paper and was in preparation when this paper appeared, and a special case was proved by
\textit{A. Conca} et al. [Algebr. Comb. 3, No. 5, 1011--1021 (2020; Zbl 1448.14052)].
Reviewer: Juan C. Migliore (Notre Dame)Algebraic cycles and Verra fourfolds.https://www.zbmath.org/1455.140082021-03-30T15:24:00+00:00"Laterveer, Robert"https://www.zbmath.org/authors/?q=ai:laterveer.robertSummary: This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial 1-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.Moduli of sheaves supported on curves of genus four contained in a quadric surface.https://www.zbmath.org/1455.140202021-03-30T15:24:00+00:00"Maican, Mario"https://www.zbmath.org/authors/?q=ai:maican.marioSummary: We study the moduli space of stable sheaves of Euler characteristic one, supported on curves of degree \((3, 3)\) contained in a smooth quadric surface. We compute its Hodge numbers by investigating the variation of the moduli spaces of \(\alpha \)-semi-stable coherent systems. We find the classification of the semi-stable sheaves by means of extensions or resolutions.Counting points on hyperelliptic curves of type \(y^2=x^{2g+1}+ax^{g+1}+bx\).https://www.zbmath.org/1455.111682021-03-30T15:24:00+00:00"Novoselov, S. A."https://www.zbmath.org/authors/?q=ai:novoselov.s-aSummary: In this work, we investigate hyperelliptic curves of type \(C:y^2=x^{2g+1}+ax^{g+1}+bx\) over the finite field \(\mathbb{F}_q,q=p^n\), \(p>2\). For the case of \(g=3\) we propose an algorithm to compute the number of points on the Jacobian of the curve with complexity \(\widetilde{O}(\log^4p)\) over \(\mathbb{F}_p\). In case of \(g=4\) we present a point counting algorithm with complexity \(\widetilde{O}(\log^8q)\) over \(\mathbb{F}_q\). The Jacobian \(J_C\) splits over an extension of the field \(\mathbb{F}_q\) on the Jacobians of the curves defined by the Dickson polynomials \(D_g(x, 1)\) of degree \(g\). For these curves of genus \(2, 3, 5\) with equation \(y^2=D_g(x,1)+a\) and curves of genus \(2, 4\) with equation \(y^2=(x+2)(D_g(x,1)+a)\), we give the lists of possible characteristic polynomials of the Frobenius endomorphism modulo \(p\).On the vanishing of relative negative \(K\)-theory.https://www.zbmath.org/1455.140162021-03-30T15:24:00+00:00"Sadhu, Vivek"https://www.zbmath.org/authors/?q=ai:sadhu.vivekDel Pezzo surfaces over finite fields.https://www.zbmath.org/1455.140462021-03-30T15:24:00+00:00"Trepalin, Andrey"https://www.zbmath.org/authors/?q=ai:trepalin.andrey-sSummary: Let \(X\) be a del Pezzo surface of degree 2 or greater over a finite field \(\mathbb{F}_q\). The image \(\Gamma\) of the Galois group \(\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q)\) in the group \(\operatorname{Aut}(\operatorname{Pic}(\overline{X}))\) is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of \(\Gamma\) in the subgroup of \(\operatorname{Aut}(\operatorname{Pic}(\overline{X}))\) preserving the anticanonical class and the intersection form is a natural invariant of \(X\). We say that the conjugacy class of \(\Gamma\) in \(\operatorname{Aut}(\operatorname{Pic}(\overline{X}))\) is the \textit{type} of a del Pezzo surface. In this paper we study which types of del Pezzo surfaces of degree 2 or greater can be realized for given \(q\). We collect known results about this problem and fill the gaps.On the Hilbert function of general fat points in \(\mathbb{P}^1\times\mathbb{P}^1\).https://www.zbmath.org/1455.140112021-03-30T15:24:00+00:00"Carlini, Enrico"https://www.zbmath.org/authors/?q=ai:carlini.enrico"Catalisano, Maria Virginia"https://www.zbmath.org/authors/?q=ai:catalisano.maria-virginia"Oneto, Alessandro"https://www.zbmath.org/authors/?q=ai:oneto.alessandroIn the paper under review the authors study the so-called interpolation problem for fat point schemes in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). A very natural motivation for this study is the celebrated Alexander-Hirschowitz theorem which provides us a complete classification of ideals of double points in general position in the \(n\)-dimensional complex projective space. In particular, this classification contains all the cases when double points fail to impose independent conditions on hypersurfaces of some degree.
Let \(S = \mathbb{C}[x_{0},x_{1},y_{0},y_{1}] = \bigoplus_{i,j} S_{i,j}\) be the bigraded coordinate ring of \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). Let \(\mathcal{P} = \{P_{1}, \dots, P_{s}\}\) be a set of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) in general position and for each point \(P_{i}\) we denote by \(\mathfrak{p}_{i}\) the prime ideal defining that point. The scheme of fat points of multiplicity \(m\geq 1\) with the support at \(\mathcal{P}\) is the scheme \(\mathbb{X}\) defined by the ideal \(I_{\mathbb{X}} = \mathfrak{p}_{1}^{m} \cap \dots \cap \mathfrak{p}_{s}^{m}\). Now for any bihomogeneous ideal \(I\) in \(S\), we define the Hilbert function of \(S/I\) as
\[\mathrm{HF}_{S/I}(a,b) : = \dim_{\mathbb{C}}(S/I)_{(a,b)} = \dim_{\mathbb{C}}S_{(a,b)} -\dim_{\mathbb{C}} I_{(a,b)} \quad \text{for} \quad (a,b) \in \mathbb{N}^{2}.\]
For short, we denote by \(\mathrm{HF}_{\mathbb{X}}\) be the Hilbert function of the quotient ring \(S / I_{\mathbb{X}}\).
The key question which gives a motivation for the paper under review can be formulated as follows.
Question: Let \(\mathbb{X}\) be a scheme of fat points of multiplicity \(m\) in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). What is the bigraded Hilbert function of \(\mathbb{X}\)?
The main contribution of the present paper is the following general result.
Theorem A. Let \(a\geq b\) and we assume that \(b\geq m\). Consider the fat point scheme \(\mathbb{X} = mP_{1} +\dots+ mP_{s} \subset \mathbb{P}^{1} \times \mathbb{P}^{1}\). Then
\[\mathrm{HF}_{\mathbb{X}}(a,b) =\min \bigg\{ (a+1)(b+1), s\binom{m+1}{2} -s\binom{m-b}{2}\bigg\},\]
except if \(s = 2k+1\) and \(a=bk+c+s(m-b)\) with \(c = 0,\dots, b-2\), where
\[\mathrm{HF}_{\mathbb{X}}(a,b) = (a+1)(b+1) - \binom{c+2}{2}.\]
Moreover, in the case of triple points, the authors are able to provide a complete description of the associated Hilbert functions.
Theorem B. Let \(\mathbb{X} = 3P_{1} + \dots + 3P_{s} \subset \mathbb{P}^{1} \times \mathbb{P}^{1}\). Then
\[\mathrm{HF}_{\mathbb{X}}(a,b) =\min\{(a+1)(b+1),6s\},\]
except for the following situations:
1) \(b=1\) and \(s < \frac{2}{5}(a+1)\), where \(\mathrm{HF}_{\mathbb{X}}(a,1) = 5s\);
2) \(s=2k+1\) and
i) \((a,b) =(4k+1,2)\), where \(\mathrm{HF}_{\mathbb{X}}(4k+1,2) = (a+1)(b+1) - 1\);
ii) \((a,b) = (3k,3)\), where \(\mathrm{HF}_{\mathbb{X}}(3k,3) = (a+1)(b+1)-1\);
iii) \((a,b) = (3k+1,3)\), where \(\mathrm{HF}_{\mathbb{X}}(3k+1,3) = 6s-1\);
3) \(s=5\) and \((a,b)=(5,4)\), where \(\mathrm{HF}_{\mathbb{X}}(5,4)=29\).
Reviewer: Piotr Pokora (Kraków)Non-abelian groups acting on Severi-Brauer surfaces.https://www.zbmath.org/1455.140922021-03-30T15:24:00+00:00"Shramov, K. A."https://www.zbmath.org/authors/?q=ai:shramov.konstantin-a(no abstract)Categorification of Legendrian knots.https://www.zbmath.org/1455.140382021-03-30T15:24:00+00:00"Kuwagaki, Tatsuki"https://www.zbmath.org/authors/?q=ai:kuwagaki.tatsukiSummary: The concept of a perverse schober defined by Kapranov-Schechtman is a categorification of the notion of a perverse sheaf. In their definition, a key ingredient is a certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of the above story, as categorification of Legendrian points/knots. The notion turns out to include various notions such as semi-orthogonal decomposition, mutation braiding, spherical functor, \(N\)-spherical functor, and irregular perverse schober.Noether's problem and rationality problem for multiplicative invariant fields: a survey.https://www.zbmath.org/1455.120042021-03-30T15:24:00+00:00"Hoshi, Akinari"https://www.zbmath.org/authors/?q=ai:hoshi.akinariSummary: In this paper, we give a brief survey of recent developments on Noether's problem and rationality problem for multiplicative invariant fields including author's recent papers [Proc. Japan Acad., Ser. A 91, No. 3, 39--44 (2015; Zbl 1334.12007)] about Noether's problem over \(\mathbb{Q}\), the author et al. [Asian J. Math. 17, No. 4, 689--714 (2013; Zbl 1291.13012)], \textit{H. Chu} et al. [J. Algebra 442, 233--259 (2015; Zbl 1327.13027); the author, J. Algebra 445, 394--432 (2016; Zbl 1368.12003)] and the author et al. [J. Algebra 458, 120--133 (2016; Zbl 1348.14032)] about Noether's problem over \(\mathbb{C}\), and the author et al. [J. Algebra 403, 363--400 (2014; Zbl 1308.12005)] and [the author et al., arXiv:1609.04142)] about rationality problem for multiplicative invariant fields.Anabelian geometry of curves over algebraically closed fields of positive characteristic: the case of one-punctured elliptic curves.https://www.zbmath.org/1455.140612021-03-30T15:24:00+00:00"Sarashina, Akira"https://www.zbmath.org/authors/?q=ai:sarashina.akiraSummary: This article is an announcement of the author's recent work on anabelian geometry over algebraically closed fields of positive characteristic. We review some known results in this area and give a sketch of the proof of the main result which concerns reconstruction of curves of \((1,1)\)-type by their geometric fundamental groups.Geometric version of the Grothendieck conjecture for universal curves over Hurwitz stacks: a research announcement.https://www.zbmath.org/1455.140622021-03-30T15:24:00+00:00"Tsujimura, Shota"https://www.zbmath.org/authors/?q=ai:tsujimura.shotaSummary: Hurwitz stacks are algebraic stacks that parametrize simple coverings. In this paper, we introduce a certain geometric version of the Grothendieck conjecture for universal curves over Hurwitz stacks. This result generalizes a similar result obtained by Hoshi and Mochizuki in the case of universal curves over moduli stacks of pointed smooth curves. After introducing these results, we give a sketch of the proof of the above version of the Grothendieck conjecture in the hyperelliptic case.Reconstruction of function fields from pro-\(p\) second configuration space groups, a research announcement.https://www.zbmath.org/1455.140592021-03-30T15:24:00+00:00"Higashiyama, Kazumi"https://www.zbmath.org/authors/?q=ai:higashiyama.kazumiSummary: In the present article, the author gives an outline of the paper ``The semi-absolute anabelian geometry of geometrically pro-\(p\) arithmetic fundamental groups of associated low-dimensional configuration spaces''. The details of the arguments appearing in the present paper may be found in [\textit{K. Higashiyama}, ``The semi-absolute anabelian geometry of geometrically pro-\(p\) arithmetic fundamental groups of associated low-dimensional configuration spaces'', RIMS preprint 1906 (August 2019)].On a direct product decomposition related to the Grothendieck-Teichmüller group.https://www.zbmath.org/1455.140602021-03-30T15:24:00+00:00"Minamide, Arata"https://www.zbmath.org/authors/?q=ai:minamide.arataSummary: In this article, we announce our results on the Grothendieck-Teichmüller group GT obtained in the joint work with Yuichiro Hoshi and Shinichi Mochizuki. Let \(n\ge 2\) be a positive integer and \(k\) an algebraically closed field of characteristic zero. Write \(\Pi_n\) for the étale fundamental group of the \(n\)-th configuration space of \(\mathbb{P}^1_k\setminus\{0,1,\infty\}\) and \(\mathfrak{S}_{n+3}\) for the symmetric group on \(n+3\) letters. Then our main theorem asserts that we have a direct product decomposition \(\text{Out}(\Pi_n)= \text{GT}\times\mathfrak{S}_{n+3}\). The detail of the arguments appearing in this article may be found in [\textit{Y. Hoshi} et al., ``Group-theoreticity of Numerical Invariants and Distinguished Subgroups of Configuration Space Groups'', RIMS Preprint 1970 (2017)].Rationally connected rational double covers of primitive Fano varieties.https://www.zbmath.org/1455.140252021-03-30T15:24:00+00:00"Pukhlikov, Aleksandr V."https://www.zbmath.org/authors/?q=ai:pukhlikov.aleksandr-vThe main result of this manuscript has as a corollary an interesting result on unirationality: A general hypersurface \(V\subset \mathbb{P}^M\) of degree \(M+1\) is not unirational, that is, there is no rational dominant map \(\mathbb{P}^M\dashrightarrow V.\)
Indeed, the author proves that if \(V\) is a primitive Fano variety satisfying suitable conditions then there no rational Galois covers \(X\stackrel{d:1}{\dashrightarrow} V\) with an abelian Galois group of order \(d\geq 2,\) where \(X\) is a rationally connected variety. Such conditions are satisfied for instance for general hypersurfaces \(V\subset \mathbb{P}^M\) of degree \(M+1.\)
Reviewer: Rick Rischter (Itajubá)Perfect squares representing the number of rational points on elliptic curves over finite field extensions.https://www.zbmath.org/1455.110532021-03-30T15:24:00+00:00"Chim, Kwok Chi"https://www.zbmath.org/authors/?q=ai:chim.kwok-chi"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florianSummary: Let \(q\) be a perfect power of a prime number \(p\) and \(E(\mathbb{F}_q)\) be an elliptic curve over \(\mathbb{F}_q\) given by the equation \(y^2=x^3+Ax+B\). For a positive integer \(n\) we denote by \(\# E(\mathbb{F}_{q^n})\) the number of rational points on \(E\) (including infinity) over the extension \(\mathbb{F}_{q^n}\). Under a mild technical condition, we show that the sequence \(\{\# E(\mathbb{F}_{q^n})\}_{n>0}\) contains at most \(10^{200}\) perfect squares. If the mild condition is not satisfied, then \(\# E(\mathbb{F}_{q^n})\) is a perfect square for infinitely many \(n\) including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range \(q<50\) and \(n\leq 1000\).On geometric complexity theory: multiplicity obstructions are stronger than occurrence obstructions.https://www.zbmath.org/1455.680692021-03-30T15:24:00+00:00"Dörfler, Julian"https://www.zbmath.org/authors/?q=ai:dorfler.julian"Ikenmeyer, Christian"https://www.zbmath.org/authors/?q=ai:ikenmeyer.christian"Panova, Greta"https://www.zbmath.org/authors/?q=ai:panova.gretaThe anabelian geometry of curves over algebraically closed fields of positive characteristic: a survey.https://www.zbmath.org/1455.140512021-03-30T15:24:00+00:00"Yang, Yu"https://www.zbmath.org/authors/?q=ai:yang.yuSummary: In the present paper, we overview some recent developments in the anabelian geometry of curves over algebraically closed fields of characteristic \(p>0\).On the arithmetic of \(K3\) surfaces with complex multiplication and its applications.https://www.zbmath.org/1455.140762021-03-30T15:24:00+00:00"Ito, Kazuhiro"https://www.zbmath.org/authors/?q=ai:ito.kazuhiroSummary: This survey article is an outline of author's talk at the RIMS Workshop Algebraic Number Theory and Related Topics (2017). We study arithmetic properties of \(K3\) surfaces with complex multiplication (CM) generalizing the results of Shimada for \(K3\) surfaces with Picard number 20. Then, following Taelman's strategy and using Matsumoto's good reduction criterion for \(K3\) surfaces with CM, we construct \(K3\) surfaces over finite fields with given \(L\)-function, up to finite extensions of the base fields. We also prove the Tate conjecture for self-products of \(K3\) surfaces over finite fields by CM lifts and the Hodge conjecture for self-products of \(K3\) surfaces with CM proved by Mukai and Buskin.Semi-algebraic properties of Minkowski sums of a twisted cubic segment.https://www.zbmath.org/1455.141112021-03-30T15:24:00+00:00"Bik, Arthur"https://www.zbmath.org/authors/?q=ai:bik.arthur"Czapliński, Adam"https://www.zbmath.org/authors/?q=ai:czaplinski.adam"Wageringel, Markus"https://www.zbmath.org/authors/?q=ai:wageringel.markusSummary: We find a semi-algebraic description of the Minkowski sum \(\mathcal{A}_{3,n}\) of \(n\) copies of the twisted cubic segment \(\{(t,t^2,t^3)\mid -1\le t\le 1\}\) for each integer \(n\ge 3\). These descriptions provide efficient membership tests for the sets \(\mathcal{A}_{3,n}\). These membership tests in turn can be used to resolve some instances of the underdetermined matrix moment problem, which was formulated by \textit{M. Rubinstein} and \textit{P. Sarnak} [``The underdetermined matrix moment Problem I'', in preparation] in order to study problems related to \(L\)-functions and their zeros.Two-graphs and the embedded topology of smooth quartics and its bitangent lines.https://www.zbmath.org/1455.140582021-03-30T15:24:00+00:00"Bannai, Shinzo"https://www.zbmath.org/authors/?q=ai:bannai.shinzo"Ohno, Momoko"https://www.zbmath.org/authors/?q=ai:ohno.momokoSummary: In this paper, we study how to distinguish the embedded topology of a smooth quartic and its bitangent lines. In order to do this, we introduce the concept of two-graphs and switching classes from graph theory. This new method improves previous results about a quartic and three bitangent lines considered by E. Artal Bartolo and J. Vallès, four bitangent lines considered by the authors and H. Tokunaga, and enables us to distinguish the embedded topology of a smooth quartic and five or more bitangent lines. As an application, we obtain a new Zariski 5-tuple and a Zariski 9-tuple for arrangements consisting of a smooth quartic and five of its bitangent lines and six of its bitangent lines, respectively.Mathieu-Zhao spaces and the Jacobian conjecture.https://www.zbmath.org/1455.141222021-03-30T15:24:00+00:00"van den Essen, Arno"https://www.zbmath.org/authors/?q=ai:van-den-essen.arnoSummary: In this paper we define the notion of a Mathieu-Zhao space, give various examples of this concept and use the framework of these Mathieu-Zhao spaces to describe a chain of challenging conjectures, all implying the Jacobian Conjecture.
For the entire collection see [Zbl 07190056].Tango structures on curves in characteristic 2.https://www.zbmath.org/1455.140692021-03-30T15:24:00+00:00"Takeda, Yoshifumi"https://www.zbmath.org/authors/?q=ai:takeda.yoshifumiSummary: The (pre-)Tango structure is a certain ample invertible sheaf of exact differential 1-forms on a projective algebraic variety and it implies some typical pathological phenomena in positive characteristic. Moreover, by using the notion of (pre-)Tango structure, we can construct another variety accompanied by similar pathological phenomena. In this article, we explicitly show several interesting and mysterious phenomena on the induced uniruled surfaces from (pre-)Tango structures on curves in characteristic 2.
For the entire collection see [Zbl 07190056].Variations on the theme of Zariski's cancellation problem.https://www.zbmath.org/1455.141212021-03-30T15:24:00+00:00"Popov, Vladimir L."https://www.zbmath.org/authors/?q=ai:popov.vladimir-leonidovichSummary: This is an expanded version of the talk by the author at the conference ``Polynomial Rings and Affine Algebraic Geometry'', February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.
For the entire collection see [Zbl 07190056].\(O_2(\mathbb{C})\)-vector bundles and equivariant real circle actions.https://www.zbmath.org/1455.140912021-03-30T15:24:00+00:00"Moser-Jauslin, L."https://www.zbmath.org/authors/?q=ai:moser-jauslin.lucySummary: The main goal of this article is to give an expository overview of some new results on real circle actions on affine four-space and their relation to previous results on \(O_2(\mathbb{C})\)-equivariant vector bundles. In [Épijournal de Géom. Algébr., EPIGA 3, Article No. 1, 11 p. (2019; Zbl 1439.14178)], we described infinite families of equivariant real circle actions on affine four-space. In the present note, we will describe how these examples were constructed, and some consequences of these results.
For the entire collection see [Zbl 07190056].Tropical combinatorial Nullstellensatz and sparse polynomials.https://www.zbmath.org/1455.141262021-03-30T15:24:00+00:00"Grigoriev, Dima"https://www.zbmath.org/authors/?q=ai:grigorev.dimitri-yu"Podolskii, Vladimir V."https://www.zbmath.org/authors/?q=ai:podolskii.vladimir-vSummary: Tropical algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects computationally accessible. Tropical polynomials play a fundamental role in this, especially for the case of algebraic geometry. On the other hand, many algebraic questions behind tropical polynomials remain open. In this paper, we address four basic questions on tropical polynomials closely related to their computational properties:
\begin{itemize}
\item [1.] Given a polynomial with a certain support (set of monomials) and a (finite) set of inputs, when is it possible for the polynomial to vanish on all these inputs?
\item [2.] A more precise question, given a polynomial with a certain support and a (finite) set of inputs, how many roots can this polynomial have on this set of inputs?
\item [3.] Given an integer \(k\), for which \(s\) there is a set of \(s\) inputs such that any nonzero polynomial with at most \(k\) monomials has a non-root among these inputs?
\item [4.] How many integer roots can have a one variable polynomial given by a tropical algebraic circuit?
\end{itemize}
In the classical algebra well-known results in the direction of these questions are Combinatorial Nullstellensatz due to N. Alon, J. Schwartz-R. Zippel Lemma and Universal Testing Set for sparse polynomials, respectively. The classical analog of the last question is known as \(\tau \)-conjecture due to M. Shub-S. Smale. In this paper, we provide results on these four questions for tropical polynomials.A graded domain is determined at its vertex. Applications to invariant theory.https://www.zbmath.org/1455.141202021-03-30T15:24:00+00:00"Gurjar, R. V."https://www.zbmath.org/authors/?q=ai:gurjar.rajendra-vasantSummary: We will prove that a positively graded domain \(/\mathbb{C}\) is uniquely determined by its completion at the irrelevant maximal ideal. As an application we will prove that the logarithmic Kodaira dimension of the smooth locus of a quotient of an affine space modulo a reductive algebraic group is \(-\infty\).
For the entire collection see [Zbl 07190056].Affine space fibrations.https://www.zbmath.org/1455.141252021-03-30T15:24:00+00:00"Gurjar, Rajendra V."https://www.zbmath.org/authors/?q=ai:gurjar.rajendra-vasant"Masuda, Kayo"https://www.zbmath.org/authors/?q=ai:masuda.kayo"Miyanishi, Masayoshi"https://www.zbmath.org/authors/?q=ai:miyanishi.masayoshiSummary: We discuss various aspects of affine space fibrations \(f:X\rightarrow Y\) including the generic fiber, singular fibers and the case with a unipotent group action on \(X\). The generic fiber \(X_\eta\) is a form of \(\mathbb{A}^n\) defined over the function field \(k(Y)\) of the base variety. Singular fibers in the case where \(X\) is a smooth (or normal) surface or a smooth threefold have been studied, but we do not know what they look like even in the case where \(X\) is a singular surface. The propagation of properties of a given smooth fiber to nearby fibers will be studied in the equivariant case of Abhyankar-Sathaye Conjecture in dimension three. We also treat the triviality of a form of \(\mathbb{A}^n\) if it has a unipotent group action. Treated subjects are classified into the following four themes
\begin{itemize}
\item[1.] Singular fibers of \(\mathbb{A}^1\)- and \(\mathbb{P}^1\)-fibrations,
\item[2.] Equivariant Abhyankar-Sathaye Conjecture in dimension three,
\item[3.] Forms of \({\mathbb A}^3\) with unipotent group actions,
\item[4.] Cancellation problem in dimension three.
\end{itemize}
For the entire collection see [Zbl 07190056].The super-rank of a locally nilpotent derivation of a polynomial ring.https://www.zbmath.org/1455.130122021-03-30T15:24:00+00:00"Freudenburg, Gene"https://www.zbmath.org/authors/?q=ai:freudenburg.geneSummary: The super-rank of a \(k\)-derivation of a polynomial ring \(k^{[n]}\) over a field \(k\) of characteristic zero is introduced. Like rank, super-rank is invariant under conjugation, and thus gives a way to classify derivations of maximal rank \(n\). For each \(m\ge 2\), we construct a locally nilpotent derivation of \(k^{[m(m+1)]}\) with maximal super-rank \(m(m+1)\).
For the entire collection see [Zbl 07190056].Rational real algebraic models of compact differential surfaces with circle actions.https://www.zbmath.org/1455.141242021-03-30T15:24:00+00:00"Dubouloz, Adrien"https://www.zbmath.org/authors/?q=ai:dubouloz.adrien"Petitjean, Charlie"https://www.zbmath.org/authors/?q=ai:petitjean.charlieSummary: We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group \(\mathbb{S}^1\) up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle \(S^1\) admits a unique smooth rational real quasi-projective model up to \(\mathbb{S}^1\)-equivariant birational diffeomorphism.
For the entire collection see [Zbl 07190056].On the theory of Gordan-Noether on homogeneous forms with zero Hessian (Improved Version).https://www.zbmath.org/1455.141182021-03-30T15:24:00+00:00"Watanabe, Junzo"https://www.zbmath.org/authors/?q=ai:watanabe.junzo"De Bondt, Michiel"https://www.zbmath.org/authors/?q=ai:de-bondt.michielSummary: We give a detailed proof for \textit{P. Gordan} and \textit{M. Nöther}'s results in [Math. Ann. 10, 547--568 (1876; JFM 08.0064.05)]. In [Bull. Braz. Math. Soc. (N.S.) 35, No. 1, 71--82 (2004; Zbl 1061.12002)], \textit{C. Lossen} has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen's paper and includes a considerable number of new observations.
For the entire collection see [Zbl 07190056].Locally nilpotent sets of derivations.https://www.zbmath.org/1455.141232021-03-30T15:24:00+00:00"Daigle, Daniel"https://www.zbmath.org/authors/?q=ai:daigle.danielSummary: Let \(B\) be an algebra over a field \(\mathbf{k}\). We define what it means for a subset of \(\mathrm{Der}_{\mathbf{k}}(B)\) to be a \textit{locally nilpotent set}. We prove some basic results about that notion and explore the following questions. Let \(L\) be a Lie subalgebra of \(\mathrm{Der}_{\mathbf{k}}(B)\); if \(L\subseteq\mathrm{LND}(B)\) then does it follow that \(L\) is a locally nilpotent set? Does it follow that \(L\) is a nilpotent Lie algebra?
For the entire collection see [Zbl 07190056].On Fano schemes of complete intersections.https://www.zbmath.org/1455.140942021-03-30T15:24:00+00:00"Ciliberto, C."https://www.zbmath.org/authors/?q=ai:ciliberto.ciro"Zaidenberg, Mikhail"https://www.zbmath.org/authors/?q=ai:zaidenberg.mikhail-gSummary: We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain projective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.
For the entire collection see [Zbl 07190056].Additive higher Chow group and module of absolute Kähler differentials: a research announcement.https://www.zbmath.org/1455.140092021-03-30T15:24:00+00:00"Onoda, Mirai"https://www.zbmath.org/authors/?q=ai:onoda.miraiSummary: This is an announcement of the original paper [the author, Kodai Math. J. 41, No. 1, 167--200 (2018; Zbl 1400.14023)]. In this paper, we give an overview about a map from the additive higher Chow group of 1-cycles to the module of absolute Kähler differentials, called a regulator map.Cotilting sheaves over weighted noncommutative regular projective curves.https://www.zbmath.org/1455.140062021-03-30T15:24:00+00:00"Kussin, Dirk"https://www.zbmath.org/authors/?q=ai:kussin.dirk"Laking, Rosanna"https://www.zbmath.org/authors/?q=ai:laking.rosannaSummary: We consider the category \(\mathrm{Qcoh}\,\mathbb{X}\) of quasicoherent sheaves where \(\mathbb{X}\) is a weighted noncommutative regular projective curve over a field \(k\). This category is a hereditary, locally noetherian Grothendieck category. We classify all indecomposable pure-injective sheaves and all cotilting sheaves of slope \(\infty \). In the cases of nonnegative orbifold Euler characteristic this leads to a classification of pure-injective indecomposable sheaves and a description of all large cotilting sheaves in \(\mathrm{Qcoh}\,\mathbb{X}\).Nontrivial linear projections on the Grassmannian \(\text{Gr}_3(\mathbb{C}^6)\).https://www.zbmath.org/1455.140972021-03-30T15:24:00+00:00"Huang, Yanhe"https://www.zbmath.org/authors/?q=ai:huang.yanhe"Petroulakis, George"https://www.zbmath.org/authors/?q=ai:petroulakis.george"Sottile, Frank"https://www.zbmath.org/authors/?q=ai:sottile.frank"Zelenko, Igor"https://www.zbmath.org/authors/?q=ai:zelenko.igorSummary: Typically, a linear projection of the Grassmannian in its Plücker embedding is generically injective, unless the image of the Grassmannian is a linear space. A notable exception are self-adjoint linear projections, which have even degree. We consider linear projections of \(\text{Gr}_3 \mathbb{C}^6\) with low-dimensional centers of projection. When the center has dimension less than five, we show that the projection has degree 1. When the center has dimension five and the projection has degree greater than 1, we show that it is self-adjoint.Arithmetic geometry of logarithmic pairs and hyperbolicity of moduli spaces. Hyperbolicity in Montréal. Based on three workshops, Montréal, Canada, 2018--2019.https://www.zbmath.org/1455.140042021-03-30T15:24:00+00:00"Nicole, Marc-Hubert (ed.)"https://www.zbmath.org/authors/?q=ai:nicole.marc-hubertPublisher's description: This textbook introduces exciting new developments and cutting-edge results on the theme of hyperbolicity. Written by leading experts in their respective fields, the chapters stem from mini-courses given alongside three workshops that took place in Montréal between 2018 and 2019. Each chapter is self-contained, including an overview of preliminaries for each respective topic. This approach captures the spirit of the original lectures, which prepared graduate students and those new to the field for the technical talks in the program. The four chapters turn the spotlight on the following pivotal themes:
\begin{itemize}
\item The basic notions of o-minimal geometry, which build to the proof of the Ax-Schanuel conjecture for variations of Hodge structures;
\item A broad introduction to the theory of orbifold pairs and Campana's conjectures, with a special emphasis on the arithmetic perspective;
\item A systematic presentation and comparison between different notions of hyperbolicity, as an introduction to the Lang-Vojta conjectures in the projective case;
\item An exploration of hyperbolicity and the Lang-Vojta conjectures in the general case of quasi-projective varieties.
\end{itemize}
Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces is an ideal resource for graduate students and researchers in number theory, complex algebraic geometry, and arithmetic geometry. A basic course in algebraic geometry is assumed, along with some familiarity with the vocabulary of algebraic number theory.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Bakker, Benjamin; Tsimerman, Jacob}, Lectures on the Ax-Schanuel conjecture, 1-68 [Zbl 1452.14007]
\textit{Campana, Frédéric}, Arithmetic aspects of orbifold pairs, 69-133 [Zbl 07300481]
\textit{Javanpeykar, Ariyan}, The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties, 135-196 [Zbl 1452.14017]
\textit{Ascher, Kenneth; Turchet, Amos}, Hyperbolicity of varieties of log general type, 197-247 [Zbl 07300483]Quadratic points of surfaces in projective 3-space.https://www.zbmath.org/1455.530362021-03-30T15:24:00+00:00"Craizer, Marcos"https://www.zbmath.org/authors/?q=ai:craizer.marcos"Garcia, Ronaldo A."https://www.zbmath.org/authors/?q=ai:garcia.ronaldo-aSummary: Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is \(\pm 1/3\), while the index of the line field is \(\pm 1\). Moreover, for an elliptic quadratic point whose cubic form is semi-homogeneous, we can use Loewner's conjecture to show that the index is at most 1. From the above local results, we can conclude some global results: A generic compact elliptic surface has at least 6 quadratic points, a compact elliptic surface with semi-homogeneous cubic forms has at least 2 quadratic points and the number of quadratic points in a hyperbolic disc is odd. By studying the behavior of the cubic form in a neighborhood of the parabolic curve, we also obtain a relation between the indices of the quadratic points of a generic surface with non-empty elliptic and hyperbolic regions.Magnetized Kepler manifolds and their quantization.https://www.zbmath.org/1455.810482021-03-30T15:24:00+00:00"Meng, Guowu"https://www.zbmath.org/authors/?q=ai:meng.guowuSummary: Let \(k\geq 1\) be an integer and \(\mu\) be the half of a \textit{nonzero} integer. It is reported here that the phase space of the \(( 2 k + 1 )\)-dimensional magnetized Kepler problem with magnetic charge \(\mu\) can be identified with the elliptic co-adjoint orbit of the real Lie algebra \(\mathfrak{so}(2,2k+2)\) that corresponds to the dominant weight \((\underbrace{|\mu|,\ldots,|\mu|}_{k+1},\mu)\).
As a corollary, one has the following result: the aforementioned elliptic co-adjoint orbit admits a natural quantization in which the Hilbert space of quantum states is the space of square integrable sections of a Hermitian vector bundle over the punctured Euclidean space in dimension \(2k+1\); moreover, this Hilbert space provides a geometric realization for the unitary highest weight \(\mathfrak{so}(2,2k+2)\)-module with highest weight
\[
(-k-|\mu|,\underbrace{|\mu|,\ldots,|\mu|}_k,\mu).
\]Links of sandwiched surface singularities and self-similarity.https://www.zbmath.org/1455.140072021-03-30T15:24:00+00:00"Fantini, Lorenzo"https://www.zbmath.org/authors/?q=ai:fantini.lorenzo"Favre, Charles"https://www.zbmath.org/authors/?q=ai:favre.charles"Ruggiero, Matteo"https://www.zbmath.org/authors/?q=ai:ruggiero.matteoThis paper studies sandwiched singularities in arbitrary characteristic. A normal 2-dimensional complete local ring \(\mathcal O\) (with algebraically closed residue field \(k\)) is \textit{sandwiched} if there exists a proper birational morphism \(X \to X_0\) from a normal algebraic surface \(X\) to a smooth surface \(X_0\) and a point \(x \in X\) such that \(\mathcal{O}\cong\widehat{\mathcal{O}_{X,x}}\). A normal point of an algebraic surface is a sandwiched singularity if its complete local ring is sandwiched. Over \(\mathbb C\) this definition is equivalent to that of \textit{M. Spivakovsky} [Ann. Math. (2) 131, No. 3, 411--491 (1990; Zbl 0719.14005)]. Given a point \(p\) on a smooth surface \(X_0\), let \(Y\to X_0\) be a sequence of point blowups centered above \(p\); contracting a connected divisor obtained by removing some components of the exceptional divisor gives a sandwiched singularity \((X, x)\) and any sandwiched singularity is locally étale isomorphic to such a singularity. The surface \(X\) is sandwiched between the smooth surfaces \(X_0\) and \(Y\), explaining the terminology.
The authors give six different characterisations of sandwiched singularities, in term of self-similarity properties. The last of them is the existence of a proper birational morphism of algebraic surfaces \(\pi\colon X'\to X\) such that \(\widehat{\mathcal{O}_{X',p}}\cong
\widehat{\mathcal{O}_{X,x}}\) for some point \(p\in \pi^{-1}(x)\). This property can also be formulated in terms of the dual graph of the
singularity. A connected weighted graph is self-similar if it admits a nontrivial modification (where an elementary modification corresponds to a point blowup) containing a subgraph isomorphic to the original graph.
The other characterisations involve the non-archimedean link \(\operatorname{NL}(X, x)\) of \((X, x)\). If \(k\) is endowed with the
trivial absolute value and \(X^{\text{an}}\) is the analytification in the sense of Berkovich geometry, then \(\operatorname{NL}^\varepsilon(X, x) = \{y\in X^{\text{an}}\mid \max_i |z_i (y)| = \varepsilon\}\) does not depend on the embedding nor on \( \varepsilon\in (0,1)\) and is called
the non-archimedean link of \((X, 0)\). Concretely it is is the set of semi-valuations \(v\) on \(\widehat{\mathcal{O}_{X,x}}\) that are
normalized by the condition \(\min_{f\in\mathfrak{m}}v( f ) = 1\). A detailed description of the analytic structure of \(\operatorname{NL}(X, x)\) is given, following the work of the first author [Trans. Am. Math. Soc. 370, No. 11, 7815--7859 (2018; Zbl 1423.14171)]. Self-similarity of the non-archimedean link \(\operatorname{NL}(X, x)\), and by extension of \((X, x)\), means that there exists a finite set \(T\) of points of
type \(1\) (endpoints corresponding to semi-valuations with nontrivial kernel) of \(\operatorname{NL}(X, x)\) such that \(\operatorname{NL}(X, x)\setminus T\) is isomorphic to an open subset \(U\) of \(\operatorname{NL}(X, x)\) with \(\overline U\) a proper subset of \(\operatorname{NL}(X, x)\). This condition is implied by the existence of a finite set \(T\) of points of type \(1\) such that every open
subset of \(\operatorname{NL}(X, x)\) contains an open subset isomorphic to \(\operatorname{NL}(X, x)\setminus T\).
The authors also searched for a characterization of sandwiched singularities in terms of their archimedean links. A self-similar property reminiscent of the last mentioned property was not found. Building on the first self-similarity property the authors prove that a (complex analytic) normal surface singularity is sandwiched if and only if its (archimedean) link can be realised as a real-analytic global strongly pseudoconvex 3-fold in a compact (non-Kähler) complex surface containing a global spherical shell. Here global means that the complement of the real 3-fold or of the spherical shell is connected.
Reviewer: Jan Stevens (Göteborg)Projective varieties with nonbirational linear projections and applications.https://www.zbmath.org/1455.141012021-03-30T15:24:00+00:00"Noma, Atsushi"https://www.zbmath.org/authors/?q=ai:noma.atsushiSummary: We work over an algebraically closed field of characteristic zero. The purpose of this paper is to characterize a nondegenerate projective variety \( X\) with a linear projection which induces a nonbirational map to its image. As an application, for smooth \( X\) of degree \( d\) and codimension \( e\), we prove the ``semiampleness'' of the \( (d-e+1)\)th twist of the ideal sheaf. This improves a linear bound of the regularity of smooth projective varieties by \textit{D. Bayer} and \textit{D. Mumford} [in: Computational algebraic geometry and commutative algebra. Proceedings of a conference held at Cortina, Italy, June 17-21, 1991. Cambridge: Cambridge University Press. 1--48 (1993; Zbl 0846.13017); \textit{A. Bertram} et al., J. Am. Math. Soc. 4, No. 3, 587--602 (1991; Zbl 0762.14012)], and gives an asymptotic regularity bound.A real variety with boundary and its global parameterization.https://www.zbmath.org/1455.530752021-03-30T15:24:00+00:00"Batkhin, A. B."https://www.zbmath.org/authors/?q=ai:batkhin.alexandr-bSummary: An algebraic variety in \(R^3\) is studied that plays an important role in the investigation of the normalized Ricci flow on generalized Wallach spaces related to invariant Einstein metrics. A procedure for obtaining a global parametric representation of this variety is described, which is based on the use of the intersection of this variety with the discriminant set of an auxiliary cubic polynomial as the axis of parameterization. For this purpose, elimination theory and computer algebra are used. Three different parameterization of the variety are obtained; each of them is valid for certain noncritical values of one of the parameters.Galois action on the Néron-Severi group of Dwork surfaces.https://www.zbmath.org/1455.110922021-03-30T15:24:00+00:00"Duan, Lian"https://www.zbmath.org/authors/?q=ai:duan.lianSummary: We study the Galois action attached to the Dwork surfaces \(X_\lambda : X_0^4 + X_1^4 + X_2^4 + X_3^4 - 4 \lambda X_0 X_1 X_2 X_3 = 0\) with parameter \(\lambda\) in a number field \(F\). We show that when \(X_\lambda\) has geometric Picard number 19, its Néron-Severi group \(\mathrm{NS}( \overline{X}_\lambda) \otimes \mathbf{Q}\) is a direct sum of quadratic characters. We provide two proofs of this result in our article. In particular, the geometric proof determines the conductor of each quadratic character. Our result matches with the one in [\textit{C. F. Doran} et al., Isr. J. Math. 228, No. 2, 665--705 (2018; Zbl 1403.14055)]. With this decomposition, this leads to a new proof of \textit{D. Wan}'s result [AMS/IP Stud. Adv. Math. 38, 159--184 (2006; Zbl 1116.11044)].Formality conjecture for minimal surfaces of Kodaira dimension 0.https://www.zbmath.org/1455.140752021-03-30T15:24:00+00:00"Bandiera, Ruggero"https://www.zbmath.org/authors/?q=ai:bandiera.ruggero"Manetti, Marco"https://www.zbmath.org/authors/?q=ai:manetti.marco"Meazzini, Francesco"https://www.zbmath.org/authors/?q=ai:meazzini.francescoSummary: Let \(\mathcal{F}\) be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra \(R\operatorname{Hom}(\mathcal{F,F})\) of derived endomorphisms of \(\mathcal{F}\) is formal. The proof is based on the study of equivariant \(L_{\infty}\) minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for \(K3\) surfaces) on previous results on the same subject.A unified method for setting finite non-commutative associative algebras and their properties.https://www.zbmath.org/1455.941832021-03-30T15:24:00+00:00"Moldovyan, Dmitriy"https://www.zbmath.org/authors/?q=ai:moldovyan.dmitrii-nikolaevichSummary: A unified method for defining a class of the finite non-commutative associative algebras of different even dimensions \(m\ge 6\) is proposed to extend the set of potential algebraic supports of the public-key cryptographic algorithms and protocols based on the hidden discrete logarithm problem. The introduced method sets the algebras containing a large set of the global left-sided units. A particular version of the method defines the algebras with parametrizable multiplication operation all modification of which are mutually associative. The cases \(m=6\) and \(m=10\) are detaily considered.The localization theorem for framed motivic spaces.https://www.zbmath.org/1455.140422021-03-30T15:24:00+00:00"Hoyois, Marc"https://www.zbmath.org/authors/?q=ai:hoyois.marcSummary: We prove the analog of the Morel-Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes.The life and work of John Tate.https://www.zbmath.org/1455.010322021-03-30T15:24:00+00:00"Serre, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:serre.jean-pierre(no abstract)Kummer for genus one over prime-order fields.https://www.zbmath.org/1455.941692021-03-30T15:24:00+00:00"Karati, Sabyasachi"https://www.zbmath.org/authors/?q=ai:karati.sabyasachi"Sarkar, Palash"https://www.zbmath.org/authors/?q=ai:sarkar.palashSummary: This work considers the problem of fast and secure scalar multiplication using curves of genus one defined over a field of prime order. Previous work by \textit{P. Gaudry} and \textit{D. Lubicz} in [Finite Fields Appl. 15, No. 2, 246--260 (2009; Zbl 1220.14023)] had suggested the use of the associated Kummer line to speed up scalar multiplication. In the present work, we explore this idea in detail. The first task is to obtain an elliptic curve in Legendre form which satisfies necessary security conditions such that the associated Kummer line has small parameters and a base point with small coordinates. It turns out that the ladder step on the Kummer line supports parallelism and can be implemented very efficiently in constant time using the single-instruction multiple-data (SIMD) operations available in modern processors. For the 128-bit security level, this work presents three Kummer lines denoted as \(K_1:=\mathsf{KL2519(81,20)}, K_2:=\mathsf{KL25519(82,77)}\) and \(K_3:=\mathsf{KL2663(260,139)}\) over the three primes \(2^{251}-9\), \(2^{255}-19\) and \(2^{266}-3\), respectively. Implementations of scalar multiplications for all three Kummer lines using Intel intrinsics have been done, and the code is publicly available. Timing results on the Skylake and the Haswell processors of Intel indicate that both fixed base and variable base scalar multiplications for \(K_1\) and \(K_2\) are faster than those achieved by \texttt{Sandy2x}, which is a highly optimised SIMD implementation in assembly of the well-known \texttt{Curve25519}. On Skylake, both fixed base and variable base scalar multiplications for \(K_3\) are faster than \texttt{Sandy2x}, whereas on Haswell, fixed base scalar multiplication for \(K_3\) is faster than \texttt{Sandy2x} while variable base scalar multiplication for both \(K_3\) and \texttt{Sandy2x} takes roughly the same time. In practical terms, the particular Kummer lines that are introduced in this work are serious candidates for deployment and standardisation. We further illustrate the usefulness of the proposed Kummer lines by instantiating the quotient Digital Signature Algorithm on all the three Kummer lines.
See the preliminary version in [Asiacrypt 2017, Lect. Notes Comput. Sci. 10625, 3--32 (2017; Zbl 1380.94105)].The Kuga-Satake construction under degeneration.https://www.zbmath.org/1455.140182021-03-30T15:24:00+00:00"Schreieder, Stefan"https://www.zbmath.org/authors/?q=ai:schreieder.stefan"Soldatenkov, Andrey"https://www.zbmath.org/authors/?q=ai:soldatenkov.andreySummary: We extend the Kuga-Satake construction to the case of limit mixed Hodge structures of \(K3\) type. We use this to study the geometry and Hodge theory of degenerations of Kuga-Satake abelian varieties associated with polarized variations of \(K3\) type Hodge structures over the punctured disc.Motivic cohomology of twisted flag varieties.https://www.zbmath.org/1455.140452021-03-30T15:24:00+00:00"Yagita, Nobuaki"https://www.zbmath.org/authors/?q=ai:yagita.nobuakiSummary: In this paper, we study the mod\((p)\) motivic cohomology of twisted complete flag varieties \(X\) over some restricted fields \(k\). Here we take fields \(k\) such that the Milnor \(K\)-theory \(K_n^M+2 (k) / p = 0\) for some \(n \geq 2\). For these fields, we compute the mod \((p)\) motivic cohomologies of the Rost motives \(R_n\) and the flag variety \(X\) containing \(R_2\).Towards a non-Archimedean analytic analog of the Bass-Quillen conjecture.https://www.zbmath.org/1455.140492021-03-30T15:24:00+00:00"Kerz, Moritz"https://www.zbmath.org/authors/?q=ai:kerz.moritz-c"Saito, Shuji"https://www.zbmath.org/authors/?q=ai:saito.shuji.1"Tamme, Georg"https://www.zbmath.org/authors/?q=ai:tamme.georgSummary: We suggest an analog of the Bass-Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles \(K_0\).Categorical proof of holomorphic Atiyah-Bott formula.https://www.zbmath.org/1455.180122021-03-30T15:24:00+00:00"Kondyrev, Grigory"https://www.zbmath.org/authors/?q=ai:kondyrev.grigory"Prikhodko, Artem"https://www.zbmath.org/authors/?q=ai:prikhodko.artemSummary: Given a \(2\)-commutative diagram
\[\begin{tikzcd}
X\rar["F_X"]\dar["\varphi" '] &X\dar["\varphi"] \\ Y\rar["F_Y" '] & Y
\end{tikzcd}\] in a symmetric monoidal \((\infty ,2)\)-category \(\mathscr{E}\) where \(X,Y\in \mathscr{E}\) are dualizable objects and \(\varphi\) admits a right adjoint we construct a natural morphism \(\operatorname{Tr}_{\mathscr{E}}(F_X)\longrightarrow\operatorname{Tr}_{\mathscr{E}}(F_Y)\) between the traces of \(F_X\) and \(F_Y\), respectively. We then apply this formalism to the case when \(\mathscr{E}\) is the \((\infty ,2)\)-category of \(k\)-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula).