Recent zbMATH articles in MSC 14Fhttps://www.zbmath.org/atom/cc/14F2021-04-16T16:22:00+00:00WerkzeugThe trace of the local \(\mathbb{A}^1\)-degree.https://www.zbmath.org/1456.140272021-04-16T16:22:00+00:00"Brazelton, Thomas"https://www.zbmath.org/authors/?q=ai:brazelton.thomas"Burklund, Robert"https://www.zbmath.org/authors/?q=ai:burklund.robert"McKean, Stephen"https://www.zbmath.org/authors/?q=ai:mckean.stephen"Montoro, Michael"https://www.zbmath.org/authors/?q=ai:montoro.michael"Opie, Morgan"https://www.zbmath.org/authors/?q=ai:opie.morganThe theory of \(\mathbb{A}^1\)-enumerative geometry is a relatively young topic, defined as an application of the \(\mathbb{A}^1\)-homotopy theory of \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to enumerative geometry over arbitrary base fields \(\kappa\); see [\textit{B. Williams} and \textit{K. Wickelgren}, in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 42 p. (2020; Zbl 07303335)].
In this short article, the authors study the local \(\mathbb{A}^1\)-degree, a foundational tool in \(\mathbb{A}^1\)-enumerative geometry. Analogous to the classical Brouwer degree of a continuous map \(\mathbb{R}^n\to \mathbb{R}^n\) with an isolated zero at the origin taking values in the integers, the local \(\mathbb{A}^1\)-degree is an invariant of a map \(f\colon\mathbb{A}^n_\kappa\to \mathbb{A}^n_\kappa\) at an isolated zero \(p\) taking values in the Grothendieck--Witt group \(\mathrm{GW}(\kappa)\) of the field \(\kappa\). The main theorem [Theorem 1.3] states that if the map \(\kappa\to\kappa(p)\) from the residue field of \(p\) is separable and of finite degree, then the local \(\mathbb{A}^1\)-degree of \(f\) can be computed as the image of the local \(\mathbb{A}^1\)-degree of the base change of \(f\) over \(\kappa(p)\) under the natural transfer map of Grothendieck-Witt groups. This result is somewhat inspired by Morel's original work in defining the local \(\mathbb{A}^1\)-degree [\textit{F. Morel}, in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035--1059 (2006; Zbl 1097.14014)] and more recent work by \textit{J. L. Kass} and \textit{K. Wickelgren} [Duke Math. J. 168, No. 3, 429--469 (2019; Zbl 1412.14014)]. The authors also obtain a corollary of Theorem 1.3, generalising a statement of Kass and Wickelgren [loc. cit.] relating the local \(\mathbb{A}^1\)-degree with the Scheja-Storch bilinear form.
The authors begin in Section 2 by reviewing the necessary background in \(\mathbb{A}^1\)-homotopy theory, including the construction of the local \(\mathbb{A}^1\)-degree, and some of the six functor formalism of \(\mathbb{A}^1\)-stable homotopy theory, as detailed by \textit{M. Hoyois} [Algebr. Geom. Topol. 14, No. 6, 3603--3658 (2014; Zbl 1351.14013)], for example. The six functor formalism allows for interpretations of the various maps defining the local \(\mathbb{A}^1\)-degree and the transfer of Grothendieck-Witt groups internally in \(\mathcal{SH}(\kappa)\), the stable motivic homotopy category of \(\kappa\).
In Section 3, the authors prove Theorem 1.3. To do this, a commutative diagram [Diagram (3)] is constructed in the unstable motivic homotopy category of \(\kappa\). Upon passage to the stable category \(\mathcal{SH}(\kappa)\), Diagram (3) yields Diagram (4) containing the local \(\mathbb{A}^1\)-degree as an endomorphism of the motivic sphere \(\mathbb{P}^n_\kappa/\mathbb{P}^{n-1}_\kappa\). Using the purity theorem of Morel-Voevodsky and an analysis of certain motivic Thom spaces, various arrows in Diagram (4) are recast and shown to be invertible; see Diagram (7). Theorem 1.3 now follows from Diagram (7) and the fact that \(\kappa\to \kappa(p)\) is finite and separable. As a final paragraph, the authors show the Scheja-Storch form agrees with the local \(\mathbb{A}^1\)-degree when \(\kappa\to\kappa(p)\) is separable and of finite degree, extending a previous result of Wickelgren and Kass [loc. cit.], where one assumes that \(p\) is \(\kappa\)-rational.
Reviewer: Jack Davies (Utrecht)Intersection cohomology of pure sheaf spaces using Kirwan's desingularization.https://www.zbmath.org/1456.140152021-04-16T16:22:00+00:00"Chung, Kiryong"https://www.zbmath.org/authors/?q=ai:chung.kiryong"Yoon, Youngho"https://www.zbmath.org/authors/?q=ai:yoon.younghoLet \(\mathbf{M}_n\) be the space parametrizing semi-stable sheaves \(F\) on \(\mathbb {P}^n\) with a linear resolution \[0\to\mathcal {O}_{\mathbb {P}^n}(-1)^2 \to \mathcal {O}_{\mathbb {P}^n}^2\to F\to 0.\] \(\mathbb {M}_n\) is an integral normal variety, \(\dim \mathbb {M}_n =4n-3\), which is the Simpsons compactification of twisted sheaves \(\mathcal{I}_{L,Q}(1)\), where \(Q\subset \mathbb {P}^n\) is a rank \(4\) hyperquadric and \(L\subset Q\) is a linear subspace of dimension \(n-2\). The authors computes the intersection Poincaré polynomial of \(\mathbf{M}_n\) using Kirwan's desingularization method and the relation between \(\mathbf{M}_n\), the GIT quotient of the Kroneker quiver (Kontsevich's map space \(\mathbf{K}_n\)). Then they compute the intersection Poincaré polynomial of the moduli space of pure one-dimensional sheaves on the smooth surfaces \(\mathbb {P}^2\), \(\mathbb{F}_0\) and \(\mathbb {F}_1\).
Reviewer: Edoardo Ballico (Povo)A note on the Brauer group and the Brauer-Manin set of a product.https://www.zbmath.org/1456.140252021-04-16T16:22:00+00:00"Lv, Chang"https://www.zbmath.org/authors/?q=ai:lv.changThe article under review examines Brauer groups and Brauer-Manin sets for products \(X \times Y\) of varieties over number fields. Consider the natural map
\[\Phi: \text{Br} (X) \oplus \text{Br} (Y) \rightarrow \text{Br} (X \times Y). \]
\textit{A. N. Skorobogatov} and \textit{Y. G. Zarhin} [J. Eur. Math. Soc. (JEMS) 16, No. 4, 749--769 (2014; Zbl 1295.14021)] proved that the cokernel of \(\Phi\) is finite provided that \(X\) and \(Y\) are smooth, projective, geometrically integral varieties. In the paper under review, the author generalizes this statement to the case when \(\left( X \times Y \right)(k) \neq 0 \) or \(H^3 \left( k, \overline{k}^\times\right) =0\). The proof relies on the result in the projective case and a comparison of cohomology groups of the varieties \(X\) and \(Y\) and their smooth compactifications.
Furthermore, the author proves that if \(X\) and \(Y\) are smooth, geometrically integral varieties over a number field, then the Brauer-Manin set
\(\left( X \times Y \right) \left( \mathbb{A}_k\right)^{\text{Br}(X\times Y)}\) coincides with the product of \(X \left( \mathbb{A}_k \right)^{\text{Br}(X)}\) and \(Y \left( \mathbb{A}_k \right)^{\text{Br}(Y)}\), relaxing the projectivity requirement in a similar result in [loc. cit.]. The proof uses similar methods to the proof in the projective case.
Reviewer: Charlotte Ure (Charlottesville)Birational models of moduli spaces of coherent sheaves on the projective plane.https://www.zbmath.org/1456.140162021-04-16T16:22:00+00:00"Li, Chunyi"https://www.zbmath.org/authors/?q=ai:li.chunyi"Zhao, Xiaolei"https://www.zbmath.org/authors/?q=ai:zhao.xiaoleiThe thema of the moduli spaces of sheaves on surfaces is a field of intense interest and study. In this very consistent paper one deals with coherent sheaves on the complex projective plane, in the frame of Bridgeland stability. The paper consists of 4 sections: \newline In section 1 classical results about stable sheaves on the projective plane are reviewed, mainly results of Drezet and Le Potier, and preparatory lemmas for the next sections are proved. \newline In section 2 one proves that the moduli space \({\mathcal M}^s_\sigma (w)\), where \(\sigma \) is a stability condition and \(w\) is a character in the Grothendieck group \(K({\mathbb P}^2)\) of \(D^b({\mathbb P}^2)\), ``is smooth and irreducible for generic \(\sigma \) and primitive \(w\)''.
In Section 3 one computes the last wall, and then one obtains a criterion for actual walls of \({\mathcal M}^s_\sigma (w)\) \newline In Section 4 one uses the above criterion for computing ``the nef and movable cone boundaries'' and one presents the example of the Chern character \((4,0,-15)\). \newline One has to note that the paper gives many references to existing literature on the subject, together with a discussion of the correlations to the paper under review. In this way it can be considered also a short challanging guide.
Reviewer: Nicolae Manolache (Bucureşti)Hermitian \(K\)-theory, Dedekind \(\zeta \)-functions, and quadratic forms over rings of integers in number fields.https://www.zbmath.org/1456.112212021-04-16T16:22:00+00:00"Kylling, Jonas Irgens"https://www.zbmath.org/authors/?q=ai:kylling.jonas-irgens"Röndigs, Oliver"https://www.zbmath.org/authors/?q=ai:rondigs.oliver"Østvær, Paul Arne"https://www.zbmath.org/authors/?q=ai:ostvaer.paul-arneSummary: We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian \(K\)-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind \(\zeta \)-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic \(K\)-theory and higher Witt-theory, and give a complete set of invariants for quadratic forms over rings of integers in number fields.Topological structure of spaces of stability conditions and topological Fukaya type categories.https://www.zbmath.org/1456.530722021-04-16T16:22:00+00:00"Qiu, Yu"https://www.zbmath.org/authors/?q=ai:qiu.yu|qiu.yu.1|qiu.yu.2Summary: This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories \textit{J. Woolf} [J. Lond. Math. Soc., II. Ser. 82, No. 3, 663--682 (2010; Zbl 1214.18010)], \textit{A. King} and \textit{Y. Qiu} [Adv. Math. 285, 1106--1154 (2015; Zbl 1405.16021)], \textit{Y. Qiu} [Adv. Math. 269, 220--264 (2015; Zbl 1319.18004)], \textit{N. Broomhead} et al. [J. Lond. Math. Soc., II. Ser. 93, No. 2, 273--300 (2016; Zbl 1376.16006)], \textit{Y. Qiu} and \textit{J. Woolf} [Geom. Topol. 22, No. 6, 3701--3760 (2018; Zbl 1423.18044)]. The key is to understand Happel-Reiten-Smalø tilting as tiling of cells. Second, we review topological realizations of various Fukaya type categories \textit{Y. Qiu} [Adv. Math. 269, 220--264 (2015; Zbl 1319.18004)], \textit{Y. Qiu} and \textit{Y. Zhou} [Compos. Math. 153, No. 9, 1779--1819 (2017; Zbl 1405.16024)], \textit{Y. Qiu} [Math. Ann. 365, No. 1--2, 595--633 (2016; Zbl 1378.16027), Math. Z. 288, No. 1--2, 39--53 (2018; Zbl 1442.16017)], \textit{Y. Qiu} and \textit{Y. Zhou} [Trans. Am. Math. Soc. 372, No. 1, 635--660 (2019; Zbl 1444.16013)], \textit{F. Haiden} et al. [Publ. Math., Inst. Hautes Étud. Sci. 126, 247--318 (2017; Zbl 1390.32010)], namely cluster/Calabi-Yau and derived categories from surfaces. The corresponding spaces of stability conditions are of `tame' nature and can be realized as moduli spaces of quadratic differentials due to Bridgeland-Smith and Haiden-Katzarkov-Kontsevich [\textit{T. Bridgeland} and \textit{I. Smith}, ``Quadratic differentials as stability conditions'', Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)], Haiden et al. [loc. cit.]; \textit{A. Ikeda} [Math. Ann. 367, No. 1--2, 1--49 (2017; Zbl 1361.14015)], \textit{A. King} and \textit{Y. Qiu} [Invent. Math. 220, No. 2, 479--523 (2020; Zbl 07187476)].
For the entire collection see [Zbl 1454.00056].The Galois action and cohomology of a relative homology group of Fermat curves.https://www.zbmath.org/1456.112172021-04-16T16:22:00+00:00"Davis, Rachel"https://www.zbmath.org/authors/?q=ai:davis.rachel"Pries, Rachel"https://www.zbmath.org/authors/?q=ai:pries.rachel-j"Stojanoska, Vesna"https://www.zbmath.org/authors/?q=ai:stojanoska.vesna"Wickelgren, Kirsten"https://www.zbmath.org/authors/?q=ai:wickelgren.kirsten-gLet \(p\) be a prime satisfying Vandiver's conjecture, i.e., such that \(p\) does not divide the order of \(h^+\) of the class group of \(\mathbb{Q}(\zeta+\zeta^{-1})\), where \(\zeta\) is a \(p\)-th root of unity. Let \(X\) be the degree \(p\) Fermat curve \(x^p+y^p=z^p\). Let \(U\subset X\) be the affine open given by \(z\neq 0\). Consider the closed subscheme \(Y\subset U\) defined by \(xy=0\). Let \(H_1(U,Y;\mathbb{Z}/p)\) denote the étale homology group with \(\mathbb{Z}/p \) coefficients, of the pair \((U\otimes \bar{K},Y\otimes\bar{K})\). By [\textit{G. W. Anderson}, Duke Math. J. 54, 501--561 (1987; Zbl 1370.11069)], the group \(H_1(U,Y;\mathbb{Z}/p)\) is a free rank-one \(\mathbb{Z}/p[\mu_p\times\mu_p]\)-module with generator \(\beta\). The Galois action of \(\sigma\in G_{\mathbb{Q}(\zeta)}\) is then determined by \(\sigma\beta=B_\sigma\beta\), for some \(B_\sigma\in \mathbb{Z}/p[\mu_p\times\mu_p]\). Anderson theoretically described \(B_\sigma\). In this paper, a closed form formula for \(B_\sigma\) is given. Intermediate results by the same authors [\textit{R. Davis} et al., Assoc. Women Math. Ser. 3, 57--86 (2016; Zbl 1416.11045)] about the isomorphism class of the Galois group of the field extension through the action of \(G_{\mathbb{Q}(\zeta)}\) factors, are strongly used.
The first application of this formula is that the norm of \(B_\sigma\) is \(0\) for almost all \(\sigma\). This is important in computing Galois cohomology as in Section 6 where a method for the efficient computation of the first cohomology group \(H^1(G_{\mathbb{Q}(\eta)}, H_1(U,Y;\mathbb{Z}/p))\) is given. This will eventually play a key role in understanding obstructions for rational points on Fermat curves as Ellenberg's obstruction related to the non-abelian Chabauty method.
A second application of the main formula is a proof of the fact that \(H_1(U;\mathbb{Z}/p)\) is trivialized by the product of \(\lfloor 2p/3\rfloor\) terms of the form \((B_\sigma-1)\).
Reviewer: Elisa Lorenzo García (Rennes)Multi-logarithmic differential forms on complete intersections.https://www.zbmath.org/1456.320052021-04-16T16:22:00+00:00"Aleksandrov, Alexandr G."https://www.zbmath.org/authors/?q=ai:aleksandrov.alexandr-g"Tsikh, Avgust K."https://www.zbmath.org/authors/?q=ai:tsikh.avgust-kSummary: We construct a complex \(\Omega_S^\bullet(\log C)\) of sheaves of multi-logarithmic differential forms on a complex analytic manifold \(S\) with respect to a reduced complete intersection \(C\subset S\), and define the residue map as a natural morphism from this complex onto the Barlet complex \(\omega_C^\bullet\) of regular meromorphic differential forms on \(C\). It follows then that sections of the Barlet complex can be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the residue map can be described explicitly in terms of certain integration current.Compatibility of Kisin modules for different uniformizers.https://www.zbmath.org/1456.110942021-04-16T16:22:00+00:00"Liu, Tong"https://www.zbmath.org/authors/?q=ai:liu.tong.1|liu.tongSummary: Let \(p\) be a prime and \(T\) a lattice inside a semi-stable representation \(V\). We prove that Kisin modules associated to \(T\) by selecting different uniformizers are isomorphic after tensoring a subring in \(W(R)\). As consequences, we show that several lattices inside the filtered \((\phi,N)\)-module of \(V\) constructed from Kisin modules are independent on the choice of uniformizers. Finally, we use a similar strategy to show that the Wach module can be recovered from the \((\phi,\hat{G})\)-module associated to \(T\) when \(V\) is crystalline and the base field is unramified.Book review of: A. Huber and S. Müller-Stach. Periods and Nori motives.https://www.zbmath.org/1456.000232021-04-16T16:22:00+00:00"Levine, Marc"https://www.zbmath.org/authors/?q=ai:levine.marc-nReview of [Zbl 1369.14001].Categorical mirror symmetry on cohomology for a complex genus 2 curve.https://www.zbmath.org/1456.530702021-04-16T16:22:00+00:00"Cannizzo, Catherine"https://www.zbmath.org/authors/?q=ai:cannizzo.catherineSummary: Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs \(X\) and \(Y\) such that the complex geometry on \(X\) mirrors the symplectic geometry on \(Y\). It allows one to deduce symplectic information about \(Y\) from known complex properties of \(X\). \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. \textit{M. Kontsevich} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)] conjectured that a complex invariant on \(X\) (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of \(Y\) (the Fukaya category, see [\textit{D. Auroux}, Bolyai Soc. Math. Stud. 26, 85--136 (2014; Zbl 1325.53001); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); \textit{D. McDuff} et al., Virtual fundamental cycles in symplectic topology. New York, NY: American Mathematical Society (2019)]). This is known as homological mirror symmetry. In this project, we first use the construction of ``generalized SYZ mirrors'' for hypersurfaces in toric varieties following \textit{M. Abouzaid} et al. [Publ. Math., Inst. Hautes Étud. Sci. 123, 199--282 (2016; Zbl 1368.14056)], in order to obtain \(X\) and \(Y\) as manifolds. The complex manifold is the genus 2 curve \(\Sigma_2\) (so of general type \(c_1 < 0\)) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model \((Y, v_0)\) equipped with a holomorphic function \(v_0 : Y \to \mathbb{C}\) which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of \(D^b Coh(\Sigma_2)\) into a cohomological Fukaya-Seidel category of \(Y\) as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{M. Abouzaid} and \textit{P. Seidel}, ``Lefschetz fibration methods in wrapped Floer cohomology'', in preparation].Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors.https://www.zbmath.org/1456.140262021-04-16T16:22:00+00:00"Antei, Marco"https://www.zbmath.org/authors/?q=ai:antei.marco"Biswas, Indranil"https://www.zbmath.org/authors/?q=ai:biswas.indranil"Emsalem, Michel"https://www.zbmath.org/authors/?q=ai:emsalem.michel"Tonini, Fabio"https://www.zbmath.org/authors/?q=ai:tonini.fabio"Zhang, Lei"https://www.zbmath.org/authors/?q=ai:zhang.lei.9The authors generalize the Galois correspondence and existence of Galois closure of a field to certain algebraic stacks over a field. More precisely, they introduce the notions of pseudo-properness and inflexibility of stacks. Then they consider a pseudo-proper and inflexible algebraic stack \(\mathcal X\) of finite type over a field \(k\) and an essentially finite cover \(f: {\mathcal Y}\to {\mathcal X}\). They also need some additional assumptions if \({\mathrm char} \, k>0\): either \(f\) is étale or \(\dim H^1 ({\mathcal X} , E)<\infty\) for all vector bundles \(E\). Then they show that there exists a unique (up to equivalence) finite map to the Nori fundamental gerbe \(\Pi ^{\mathrm N}_ {{\mathcal X}/k}\) of \({\mathcal X}/k\), whose base change along \({\mathcal X} \to \Pi ^{\mathrm N}_{{\mathcal X}/k}\) gives \(f\). They also prove some additional criteria on when \({{\mathcal Y}/k}\) is inflexible in case \(f\) is étale or a torsor. As a corollary they get a Galois correspondence between pointed essentially finite covers \(({\mathcal Y}, y)\to ({\mathcal X}, x)\) with inflexible \(\mathcal Y\) and subgroups of finite index in the Nori fundamental group of \(\mathcal X\).
The authors prove also existence of a Galois closure for (pointed) towers of torsors under finite group schemes over a pseudo-proper and inflexible algebraic stack of finite type over a field. They also show that previous attempts to construct such closures fail and their assumptions in positive characteristic are necessary. In particular, the construction provided in [\textit{M. A. Garuti}, Proc. Am. Math. Soc. 137, No. 11, 3575--3583 (2009; Zbl 1181.14053)] is incorrect.
Part of the paper is devoted to extension of the above results from the Nori set-up to the so called S-fundamental gerbes that are defined using numerically flat bundles.
Reviewer: Adrian Langer (Warszawa)Stability and Fourier-Mukai transform on elliptic threefolds.https://www.zbmath.org/1456.140222021-04-16T16:22:00+00:00"Lo, Jason"https://www.zbmath.org/authors/?q=ai:lo.jason|lo.jason.1Summary: This article is based on an invited talk of the same title, given at the second annual meeting of the International Consortium of Chinese Mathematicians (ICCM) in Taipei, December 2018. After a quick survey of the concepts of slope stability, Bridgeland stability and polynomial stability, the notion of a scale is introduced as an intermediary between a heart and a slicing. Drawing from the case of an elliptic curve as a motivation, we expand the use of scales to elliptic threefolds. This forms a key step in the proof of a preservation of stability result under a relative Fourier-Mukai transform on elliptic threefolds.
For the entire collection see [Zbl 1454.00057].Immaculate line bundles on toric varieties.https://www.zbmath.org/1456.140612021-04-16T16:22:00+00:00"Altmann, Klaus"https://www.zbmath.org/authors/?q=ai:altmann.klaus"Buczyński, Jarosław"https://www.zbmath.org/authors/?q=ai:buczynski.jaroslaw"Kastner, Lars"https://www.zbmath.org/authors/?q=ai:kastner.lars"Winz, Anna-Lena"https://www.zbmath.org/authors/?q=ai:winz.anna-lenaFor an algebraic variety \(X\) over an algebraically closed field \({\mathbb K}\) of arbitrary characteristic, a sheaf \({\mathcal F}\) on \(X\) is called \textit{immaculate} if all cohomology groups \(H^p(X,{\mathcal F})=0\) for all \(p\in{\mathbb Z}\). The main focus of the paper under review is the structure of the family of all immaculate line bundles on \(X\) as a subset of the group \(\text{Pic}(X)\).
Classically, the cohomology of a Weil divisor on a toric variety is calculated using polyhedra complexes contained in \(N_{\mathbb R}\), where \(N\) is the lattice of \(1\)-parameter subgroups of the torus acting on \(X\). The first contribution of the paper under review is a shift of the classical approach, viewing now the cohomology of a toric \({\mathbb Q}\)-Cartier Weil divisor using polytopes in the space \(M_{\mathbb R}\), where \(M\) is the dual lattice of \(N\). For projective toric varieties, Theorem 3.6 describes the \(M\)-graded cohomology groups \(H^i(X,{\mathcal O}(D))\) in terms of the polyhedra associated to a decomposition of the divisor \(D\) as the difference \(D^+-D^-\) of two nef divisors.
For the main objective, a description of the locus of all immaculate line bundles in the class group \(\text{Pic}(X)\) of a toric variety \(X\), the first results establish some general invariance properties of immaculacy (or a relative version of it) of locally free sheaves under various types of morphisms between toric varieties. Next, to describe the immaculate locus the authors use the map \(\pi:{\mathbb Z}^{\Sigma(1)}\to \text{Pic}(X)\) that assigns to a \(T\)-invariant divisor its class. Using this map, the first task is to identify the \(T\)-invariant divisors whose images carry some cohomology by using an approach similar to the one used for acyclic line bundles as in [\textit{L. Borisov} and \textit{Z. Hua}, Adv. Math. 221, No. 1, 277--301 (2009; Zbl 1210.14006)] and [\textit{A. I. Efimov}, J. Lond. Math. Soc., II. Ser. 90, No. 2, 350--372 (2014; Zbl 1318.14047)]. In Section 5 of the paper under review the authors identify some subsets of \(\Sigma(1)\) whose images under \(\pi\) either carry some cohomology or not. One of the main results, Theorem 5.24, essentially describes the locus of immaculate line bundles for a complete simplicial toric variety. Moreover, in some concrete instances the conditions on the subsets of \(\Sigma(1)\) can be used to describe the locus of immaculate bundles, for example for smooth projective toric varieties of Picard rank \(2\) in Theorem 6.2 . Using the classification of smooth projective toric varieties of Picard rank \(3\) of \textit{V. L. Batyrev} [Tôhoku Math. J., II. Ser. 43, No. 4, 569--585 (1991; Zbl 0792.14026)] in Section 8 the authors consider this situation in two cases, depending on the splitting of the fan of the toric variety.
Reviewer: Felipe Zaldívar (Ciudad de México)Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group.https://www.zbmath.org/1456.140642021-04-16T16:22:00+00:00"Oloo, Stephen"https://www.zbmath.org/authors/?q=ai:oloo.stephenBraden and MacPherson provided a combinatorial approach to computing equivariant intersection cohomology of (certain) varieties equipped with a torus action using the notion of moment graph [\textit{T. Braden} and \textit{R. MacPherson}, Math. Ann. 321, No. 3, 533--551 (2001; Zbl 1077.14522)]. The moment graph encodes the data of zero and one-dimensional torus orbits, and Braden and MacPherson's approach relies on the fact that it is possible to compute equivariant intersection cohomology using data from these orbits only, proved by \textit{M. Goresky} et al. [Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)]. This approach applies for example to Schubert varieties in flag varieties, which are Borel orbit closures.
In the paper under review, the author proposes a generalization of moment graphs and applies it to provide a combinatorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. The author expects applications of his approach to the representation theory of Hecke algebras.
The main difference with the previous situation is that instead of torus fixed points, the vertices of the generalized moment graph encode minimal torus orbits inside Borel orbits, which may be of arbitrary dimension. Some of the main arguments in the paper then involve working on transverse slices to such orbits, which in particular allows to describes edges of the generalized moment graph by using results of \textit{M. Brion} [Transform. Groups 4, No. 2--3, 127--156 (1999; Zbl 0953.14004)].
Following the ideas of Brendan and MacPherson, the author defines a so-called Brendan-MacPherson sheaf, of combinatorial nature, on the generalized moment graph. There is as well an intersection cohomology sheaf, directly related to intersection cohomology of the Borel orbit closures. The main result of the paper is that these two sheaves are isomorphic in an essentially canonical way, that is, the isomorphism is unique up to scalar multiplication.
Reviewer: Thibaut Delcroix (Montpellier)Representation type of surfaces in \(\mathbb{P}^3\).https://www.zbmath.org/1456.140212021-04-16T16:22:00+00:00"Ballico, Edoardo"https://www.zbmath.org/authors/?q=ai:ballico.edoardo"Huh, Sukmoon"https://www.zbmath.org/authors/?q=ai:huh.sukmoonA possible way to measure the complexity of a given \(n\)-dimensional polarized variety \((X, \mathcal{O}_X (1))\) is to ask for the families of non-isomorphic indecomposable aCM (arithmetically Cohen-Macaulay) vector bundles that it supports (recall that a vector bundle \(\mathcal{E}\) on \(X\) is aCM if \(H^i(X,\mathcal{E}\otimes\mathcal{O}_X(t))= 0\) for all \(t\in\mathbb{Z}\) and \(i=1,\dots, n-1\)). The first result on this direction was Horrocks' theorem which states that on the projective space the only indecomposable aCM bundle up to twist is the structure sheaf \(\mathcal{O}_{\mathbb{P}^n}\).
Inspired by analogous classifications in quiver theory and representation theory, a classification of polarized varieties as \textit{finite, tame and wild} was proposed. ACM varieties of finite type (namely, supporting only a finite number of non-isomorphic indecomposable aCM vector bundles) were completely classified in [\textit{ D. Eisenbud} and \textit{J. Herzog}, Math. Ann. 280, No. 2, 347--352 (1988; Zbl 0616.13011]. If we look at the other extreme of complexity we would find the varieties of wild representation type, namely, varieties for which there exist \(r\)-dimensional families of non-isomorphic indecomposable aCM bundles for arbitrary large \(r\). Recently, the representation type of any reduce aCM polarized variety has been determined [\textit{D. Faenzi} and \textit{J. Pons-Llopis}, ``The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties'', Preprint, \url{arXiv:1504.03819}].
In the article under review, the authors prove that every surface \(X\) with a regular point in the three-dimensional projective space of degree at least four is of wild representation type under the condition that either \(X\) is integral or Pic\((X)\) is \(\mathbb{Z}\)-generated by \(\mathcal{O}_X(1)\). Alongside, they also prove the interesting result that every non-integral aCM variety of dimension at least two is also very wild: namely there exist arbitrarily large dimensional families of pairwise non-isomorphic aCM non-locally free sheaves of rank one.
Reviewer: Joan Pons-Llopis (Maó)\(A_\infty \)-structures associated with pairs of 1-spherical objects and noncommutative orders over curves.https://www.zbmath.org/1456.140232021-04-16T16:22:00+00:00"Polishchuk, Alexander"https://www.zbmath.org/authors/?q=ai:polishchuk.alexander-eSummary: We show that pairs \((X,Y)\) of 1-spherical objects in \(A_\infty \)-categories, such that the morphism space \(\operatorname{Hom}(X,Y)\) is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact, we establish a more precise correspondence at the level of isomorphism of moduli spaces which we show to be affine schemes of finite type over \(\mathbb{Z} \).Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda's theorem.https://www.zbmath.org/1456.140242021-04-16T16:22:00+00:00"Basu, Saugata"https://www.zbmath.org/authors/?q=ai:basu.saugata"Patel, Deepam"https://www.zbmath.org/authors/?q=ai:patel.deepamSummary: In this article, we use cohomological techniques to obtain an algebraic version of Toda's theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. This result follows from a general `connectivity' result in cohomology. More precisely, given a closed subvariety \(X \subset \mathbb{P}^n\) over an algebraically closed field \(k\), and denoting by \(\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\dots ,\mathrm{J} (X,X)\cdots)\) the \(p\)-fold iterated join of \(X\) with itself, we prove that the restriction homomorphism on (singular or \(\ell\)-adic etale) cohomology \(\mathrm{H}^i (\mathbb{P}^N) \rightarrow \mathrm{H}^i(\mathrm{J}^{[p]}(X))\), with \(N = (p+1)(n+1)-1\), is an isomorphism for \(0 \leq i<p\), and injective for \(i=p\). We also prove this result in the more general setting of relative joins for \(X\) over a base scheme \(S\), where \(S\) is of finite type over \(k\). We give several other applications of this connectivity result including a cohomological version of classical quantifier elimination in the first order theory of algebraically closed fields of arbitrary characteristic, and to obtain effective bounds on the Betti numbers of images of projective varieties under projection maps.