Recent zbMATH articles in MSC 14Chttps://www.zbmath.org/atom/cc/14C2021-04-16T16:22:00+00:00WerkzeugComponents of the Hilbert scheme of smooth projective curves using ruled surfaces.https://www.zbmath.org/1456.140102021-04-16T16:22:00+00:00"Choi, Youngook"https://www.zbmath.org/authors/?q=ai:choi.youngook"Iliev, Hristo"https://www.zbmath.org/authors/?q=ai:iliev.hristo"Kim, Seonja"https://www.zbmath.org/authors/?q=ai:kim.seonjaLet \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth, irreducible, non-degenerate curve. \textit{L. Ein} showed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) when \(r = 3\) [Ann. Sci. Éc. Norm. Supér. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are various examples showing reducibility of \({\mathcal I}_{d,g,r}\) when \(d \geq g+r\) and \(r > 4\), disproving a claim of Severi.
Most of these examples were constructed with families of curves that are \(m\)-fold covers of \(\mathbb P^1\) with \(m \geq 3\), but the authors gave an example with a family of curves that are double covers of irrational curves [Taiwanese J. Math. 21, 583--600 (2017; Zbl 1390.14019)].
Here the authors reconstruct their example in a more geometric way as a family \(\mathcal D\) of curves on ruled surfaces. The new construction allows them to show that \(\mathcal D\) is generically smooth of expected dimension, hence a regular component. When including the distinguished component dominating \(\mathcal M_g\), this gives the first examples of Hilbert schemes \({\mathcal I}_{d,g,r}\) satisfying \(d \geq g+r\) with \textit{two} regular components.
Reviewer: Scott Nollet (Fort Worth)Birational superrigidity and \(K\)-stability of Fano complete intersections of index \(1\).https://www.zbmath.org/1456.140502021-04-16T16:22:00+00:00"Zhuang, Ziquan"https://www.zbmath.org/authors/?q=ai:zhuang.ziquanA Fano variety \(X\) is said to be birationally superrigid if it has terminal singularities,
it is \({\mathbb Q}\)-factorial of Picard number 1, and every birational map \(X\)
to a Mori fiber space is an isomorphism. On the other hand, \(X\) is \(K\)-stable
with respect to its anticanonical bundle if, essentially, it admits a Kähler-Einstein
metric, and \(K\)-stability is encoded in the positivity of the invariants \(\beta(F)\)
for \(F\) any dreamy prime divisor \(F\) over \(X\) (see 2.2 for details). In the paper under
review the author shows (see Thm. 1.2 and 1.3) that for a \(n\)-dimensional smooth Fano complete
intersection \(X \subset {\mathbb P}^{n+r}\) of index one, if \(n \geq 10r\) then \(X\) is birationally
superrigid and \(K\)-stable. Moreover, the smooth complete intersection of a quadric and a cubic
in \({\mathbb P}^5\) is also \(K\)-stable. For a Fano manifold (see Def. A.1 in the appendix of
the paper under review) \(X\) is said to be conditionally birationally superrigid if
every birational map from \(X\) to a Mori fiber space whose undefined locus has
codimension at least \(1\) plus the index of \(X\) is an isomorphism.
In the Appendix, the authors show that Fano complete intersections of higher index in large dimension (see Cor. A.3
for details) are conditionally birationally superrigid.
Reviewer: Roberto Muñoz (Madrid)Virtual classes of parabolic \(\operatorname{SL}_2(\mathbb{C})\)-character varieties.https://www.zbmath.org/1456.140652021-04-16T16:22:00+00:00"González-Prieto, Ángel"https://www.zbmath.org/authors/?q=ai:gonzalez-prieto.angelLet \(\Sigma_g\) be the closed orientable surface of genus \(g\) and \(Q\) a parabolic structure on \(\Sigma_g\). In this paper, the author completes his study of the virtual classes of the \(\operatorname{SL}_2({\mathbb C})\)-character varieties of \((\Sigma_g,Q)\) by considering the case where there are parabolic points of semi-simple type. More precisely, let \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) denote the representation variety of \((\Sigma_g,Q)\) and \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q):={\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)//\operatorname{SL}_2({\mathbb C})\) the corresponding character variety. Now let \(\operatorname{K\mathbf{Var}}_{\mathbb C}\) be the Grothendieck ring of complex algebraic varieties and \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\) the localisation of this ring with respect to the multiplicative set generated by \(q\), \(q+1\) and \(q-1\), where \(q\) is the class of \({\mathbb C}\) in \(\operatorname{K\mathbf{Var}}_{\mathbb C}\). Then the author computes explicitly the virtual class of \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) in \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\) when there is at least one parabolic point with semi-simple holonomy and possibly some additional parabolic points with holonomy of Jordan type \(J_+\) (Theorem 5.6). From this, he deduces a formula for the virtual class of \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\), valid for all holonomies (Theorem 6.1).
Character varieties have been much studied in recent years by both arithmetic and geometric methods. Both methods have limitations when there are parabolic points. In his thesis, the author developed a method involving TQFTs to avoid these limitations and used this method to compute the classes of \({\mathfrak X}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) and \({\mathcal R}_{\operatorname{SL}_2({\mathbb C})}(\Sigma_g,Q)\) in \(\operatorname{K\mathbf{MHS}}\) where the punctures are of Jordan type or type \(-\operatorname{Id}\). Here, \(\operatorname{K\mathbf{MHS}}\) is the Grothendieck ring of the category of mixed Hodge structures. (The relevant part of the author's et al. [Bull. Sci. Math. 161, Article ID 102871, 33 p. (2020; Zbl 1441.57031)]). However, there are new complications when parabolic points of semi-simple type are involved. In particular, the ``core submodule'' constructed by the author is no longer invariant under the TQFT. Moreover, if the punctures are non-generic, a new interaction phenomenon arises. These problems are addressed in the current paper.
In section 2, the author sketches the construction of the TQFT mentioned above together with a modification which allows computations in \(\operatorname{\widetilde{K}\mathbf{Var}}_{\mathbb C}\). The key section 3 is concerned with \(\operatorname{SL}_2({\mathbb C})\)-representation varieties and is preliminary to the computation of the geometric TQFT in section 4. The interaction phenomenon is described in section 5, culminating in Theorem 5.6. Section 6 is directed towards proving Theorem 6.1.
The author comments that there is much work still to be done in extending his results to groups other than \(\operatorname{SL}_2({\mathbb C})\) and to more general spaces, for example singular and non-orientable surfaces or knot complements.
Reviewer: P. E. Newstead (Liverpool)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)Algebraic cycles on moduli space of polarized hyperkähler manifolds.https://www.zbmath.org/1456.140122021-04-16T16:22:00+00:00"Li, Zhiyuan"https://www.zbmath.org/authors/?q=ai:li.zhiyuanSummary: We discuss the recent progress on studying algebraic cycle classes on the moduli space of polarized \(K3\) surfaces and more generally, polarized hyperkähler manifolds. This includes the construction of the tautological ring on these moduli spaces using Noether-Lefschetz theory and Kappa classes defined by Marian-Oprea-Pandaripande. These concepts have been recently applied to study the Chow groups and cohomology groups of these moduli spaces via various methods. In particular, we will discuss some fundamental questions and conjectures, such as the Noether-Lefschetz conjecture, tautological conjecture and generalized Franchetta conjecture. This paper is contributed to the first annual meeting of ICCM.
For the entire collection see [Zbl 1454.00056].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.On the degeneracy of integral points and entire curves in the complement of nef effective divisors.https://www.zbmath.org/1456.111192021-04-16T16:22:00+00:00"Heier, Gordon"https://www.zbmath.org/authors/?q=ai:heier.gordon"Levin, Aaron"https://www.zbmath.org/authors/?q=ai:levin.aaronSummary: As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.Homology theory formulas for generalized Riemann-Hurwitz and generalized monoidal transformations.https://www.zbmath.org/1456.570222021-04-16T16:22:00+00:00"Glazebrook, James F."https://www.zbmath.org/authors/?q=ai:glazebrook.james-f"Verjovsky, Alberto"https://www.zbmath.org/authors/?q=ai:verjovsky.albertoAuthors' abstract: In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincaré duality is defined. We further consider possible applications and examples including branched covering maps, singular varieties involving virtual tangent bundles, the Chern-Schwartz-MacPherson class, the homology L-class, generalized signature, and the cohomology signature class.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Seshadri constants of the anticanonical divisors of Fano manifolds with large index.https://www.zbmath.org/1456.140112021-04-16T16:22:00+00:00"Liu, Jie"https://www.zbmath.org/authors/?q=ai:liu.jie.7|liu.jie|liu.jie.4|liu.jie.6|liu.jie.5|liu.jie.2|liu.jie.3|liu.jie.1Let \(X\) be a normal complex projective variety, \(L\) a nef line bundle over \(X\), and \(x \in X\).
The Seshadri constant \(\epsilon(X,L;x)\) is the infimum over all irreducible cuves \(C \subset X\)
of the quotients of the \(L\)-degree of \(C\) by the multiplicity of \(C\) at \(x\). It measures the
local positivity of \(L\) at \(x\) and, as a function over \(X\), is a lower-continuous function (in the
topology whose closed subsets are countable union of Zariski closed sets). The maximum value,
attained at a very general \(x \in X\), is denoted as \(\epsilon(X,L;1)\), and is upper
bounded by \(\sqrt[n]{L^n}\). Several lower bounds are known when \(L\) ample
(see the Introduction of the paper under
review and references therein), and it is conjectured to be lower bounded by one when \(X\) smooth
and \(L\) ample (see Conj. 1.2). In the particular case of \(X\) a Fano manifold, and \(L\) the
anticanonical bunde, one can consider the question of classifying Fano manifolds for which
\(\epsilon(X,-K_X,1) \leq 1\). The list is known for Del Pezzo surfaces (see Theorem 1.4) and they are
exactly the ones for which the linear system \(|-K_X|\) is not base point free. In Theorem 1.5 the
author extends some previously known results to show that when the index \(r_X\) of the
Fano variety \(X\) is greater than or equal to \(\dim(X)-3\) then \(\epsilon(X,-K_X;1) \geq r_X\) as predicted.
In Theorem 1.6, it is shown that when the index is greater than or equal to the maximum of \(2\) and
\(\dim(X)-2\) then \(\epsilon(X,-K_X;1) =r_X\) and also equal to the minimal anticanonical degree of
a covering family of rational curves. Finally, in Theorem 1.7, a explicit computation of \(\epsilon(X,-K_X;1)\)
for smooth Fano threefolds with Picard number greater than or equal to two is provided.
This, together with previously known results lead to the corollary (see Cor. 1.8) that for \(X\) a smooth Fano threefolds very general in
its deformation family, \(\epsilon(X,-K_X;1) \leq 1\) is equivalent to the fact that \(|-K_X|\)
is not base point free. Examples in dimension \(4\) show (see Ex. 1.10) that the same result does
not hold in higher dimension but the question on the non-emptyness of
the base locus of \(|-K_X|\) for Fano manifolds such that \(\epsilon(X,-K_X;1) \leq 1\)
is posed.
Reviewer: Roberto Muñoz (Madrid)Differential forms and quadrics of the canonical image.https://www.zbmath.org/1456.140132021-04-16T16:22:00+00:00"Rizzi, Luca"https://www.zbmath.org/authors/?q=ai:rizzi.luca"Zucconi, Francesco"https://www.zbmath.org/authors/?q=ai:zucconi.francescoSummary: We extend the theory of \textit{G. P. Pirola} and \textit{F. Zucconi} [J. Algebr. Geom. 12, No. 3, 535--572 (2003; Zbl 1083.14515)]. We introduce the new notion of adjoint quadric for canonical images of irregular varieties. Using this new notion, we obtain the infinitesimal Torelli theorem for varieties whose canonical image is a complete intersection of hypersurfaces of degree \(>2\) and for Schoen surfaces. Finally, we show that a family with fiberwise liftable holomorphic forms such that the fibers have Albanese morphism of degree 1 is birationally trivial if there exist no adjoint quadrics.Feynman integrals and periods in configuration spaces.https://www.zbmath.org/1456.811922021-04-16T16:22:00+00:00"Ceyhan, Özgür"https://www.zbmath.org/authors/?q=ai:ceyhan.ozgur"Marcolli, Matilde"https://www.zbmath.org/authors/?q=ai:marcolli.matildeThe authors study Feynman amplitudes on two different configuration spaces. The first considered configuration space is the classical configuration space appearing in qunatum field theory. Whereas, the second considered configuration space can be understood as complexification of the first one.
The overall question through out the article is whether a Feynman amplitude is in the class of mixed Tate periods/motives or non-mixed Tate. To do so, the authors have used on the different configuration spaces different techniques.
For the classical configuration space, explicit calculations are performed with analytic methods commonly used in physics for configuration space computations. The Greens function is expanded in so-called Gegenbauer polynomials to separated the spherical part of the integration from the radial part. After some technical lemmata, an exact result is provided for the leading order term of the spherical integration of banana graph with three edges in four spacetime dimensions. The result consists of a restricted nested sum similar to the nested sums of multiple polylogarithms.
It is further proved that this restricted nested sum is expressible in terms of Mordell-Tornheim and Apostol-Vu multiple series. These series are in turn expressible by multiple zeta values and therefore mixed Tate periods. Consequently, the selected example and similar expressions for other Feynman amplitude are mixed Tate periods.
The Feynman amplitudes in the complexified configuration space are studied with algebro-geometric techniques. The configuration space is compactified with the so-called wonderful compactification, which is a generalization of the Fulton-MacPherson compactification. The pullback of the differential forms and of the chain of integration to the compactified space is developed. It is argued that the Feynman amplitudes are expressible as an integral of an algebraic differential form over a variety with mixed Tate motive, where the algebraic differential forms are not necessarily defined over the rationals.
Since the amplitudes need in general regularization to avoid divergencies, this is additionally done for the complex configuration space in two different ways after compactification. The first way is the current-regularization via residue current and principle value current. The second regularization method is a regularization via deformation, where the complex compactified configuration space is extended by a trivial fibration. The property of being a mixed Tate motive is for both regularizations preserved.
In summary, the article underpins the observation that a big class of Feynman amplitudes consists of mixed Tate periods/motives. Very interesting examples are considered, computed and constructed, which gives rise to further investigations for more general results. The relevance is of a good type for the mathematical physics community.
For the entire collection see [Zbl 1446.81001].
Reviewer: Alexander Hock (Münster)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].