Recent zbMATH articles in MSC 14Lhttps://www.zbmath.org/atom/cc/14L2021-04-16T16:22:00+00:00WerkzeugHomogeneous vector bundles over abelian varieties via representation theory.https://www.zbmath.org/1456.140522021-04-16T16:22:00+00:00"Brion, Michel"https://www.zbmath.org/authors/?q=ai:brion.michelLet \(A\) be an abelian variety over a field. A vector bundle on \(A\) is called homogeneous if it is invariant under pullback by translations on \(A\). Such bundles have been studied by many people, especially in the case when \(k\) is algebraically closed. The author proposes an alternative aproach to studying such bundles that works over an arbitrary field. This is based on the equivalence of the category of homogeoenus bundles with the category of finite-dimensional representations of a certain commutative affine \(k\)-group scheme \(H_A\), which plays a role of the affine fundamental group scheme of \(A\). Roughly speaking, by taking the inverse limit over all extensions \(1\to H\to G\to A\to 1\) with \(H\) affine, one arrives at the universal extension \(1\to H_A\to G_A\to A\to 1\). Then to a finite-dimensional \(H_A\)-module one can associate a homogeneous vector bundle \(G_A\times ^{H_A}V\to G_A/H_A=A\). Over an algebraically closed field \(k\) the group scheme \(H_A\) coincides with the S-fundamental group scheme studied by the reviewer.
The paper contains also various interesting results pertaining to representation theory of a commutative group scheme over a possibly imperfect field.
Reviewer: Adrian Langer (Warszawa)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)Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms.https://www.zbmath.org/1456.140452021-04-16T16:22:00+00:00"Pignatelli, Roberto"https://www.zbmath.org/authors/?q=ai:pignatelli.robertoThis paper continues the study of product--quotient surfaces, i.e., surfaces that are the quotient of a product of two algebraic curves by a finite group of automorphisms (cf. [\textit{I. Bauer} and \textit{R. Pignatelli}, Groups Geom. Dyn. 10, No. 1, 319--363 (2016; Zbl 1348.14021); \textit{D. Frapporti} and \textit{R. Pignatelli}, Glasg. Math. J. 57, No. 1, 143--165 (2015; Zbl 1330.14069); \textit{I. Bauer} et al., Am. J. Math. 134, No. 4, 993--1049 (2012; Zbl 1258.14043)].
Here the case of mixed surfaces is looked at. A mixed surface is the minimal resolution S of the singularities of a quotient \((C \times C)/G\) of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors.
The main result is a very precise description of the Albanese morphism of \(S\), when \(S\) is irregular (i.e. when \(q(S)>0\)). It is shown that \(S\) has always maximal Albanese dimension when possible (i.e. when \(q(S)\geq 2\)). Then the result is applied to all the semi-isogenous mixed surfaces with \(p_g=q=2\) constructed by \textit{N. Cancian} and \textit{D. Frapporti} [Math. Nachr. 291, No. 2--3, 264--283 (2018; Zbl 1408.14122)].
The main tool used here is studying further quotients \((C \times C)/G' \) where \(G '\) is a group of automorphisms of \(C\times C\) containing \(G\), and relating the Albanese morphism of \(S\) with the Jacobian \(J(C)\).
Reviewer: Margarida Mendes Lopes (Lisboa)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)The pluricanonical systems of a product-quotient variety.https://www.zbmath.org/1456.140582021-04-16T16:22:00+00:00"Favale, Filippo F."https://www.zbmath.org/authors/?q=ai:favale.filippo-francesco"Gleissner, Christian"https://www.zbmath.org/authors/?q=ai:gleissner.christian"Pignatelli, Roberto"https://www.zbmath.org/authors/?q=ai:pignatelli.robertoSummary: We give a method for the computation of the plurigenera of a product-quotient manifold, and two different types of applications of it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a product-quotient surface of general type.
For the entire collection see [Zbl 07237934].Equidistribution of expanding translates of curves and Diophantine approximation on matrices.https://www.zbmath.org/1456.220032021-04-16T16:22:00+00:00"Yang, Pengyu"https://www.zbmath.org/authors/?q=ai:yang.pengyuOne can begin with author's abstract:
``We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space \(G/\Gamma\) of a semisimple algebraic group \(G\). We define two families of algebraic subvarieties of the associated partial flag variety \(G/\Gamma\), which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of \(m\times n\) real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.''
It is noted that many problems in number theory can be recast in the language of homogeneous dynamics. A survey is devoted to this fact, to the main problems of this research, and to the motivation of the present investigations. Notions that are useful for proving the main statements are recalled and explained.
The main results and several auxiliary statements are proven with explanations. Applications of the main results and also connections between these results and known researches are noted.
Reviewer: Symon Serbenyuk (Kyïv)Sustained \(p\)-divisible groups and a foliation on moduli spaces of abelian varieties.https://www.zbmath.org/1456.140572021-04-16T16:22:00+00:00"Chai, Ching-Li"https://www.zbmath.org/authors/?q=ai:chai.chingliSummary: We explain a concept of sustained \(p\)-divisible groups, discovered in collaboration with Frans Oort and motivated by the Hecke orbit problem. This concept leads to a scheme-theoretic definition of central leaves in moduli spaces of abelian varieties in characteristic \(p>0\). We also formulate a notion of strongly Tate-linear formal subschemes of the sustained deformation space \(\mathbf{Def}^{\text{sus}}(Y_0)\) of a \(p\)-divisible group \(Y_0\), and a local rigidity question on whether every reduced and irreducible closed formal subscheme of \(\mathbf{Def}^{\text{sus}}(Y_0)\) stable under a strongly non-trivial action of a \(p\)-adic Lie group is strongly Tate-linear.
For the entire collection see [Zbl 1454.00057].Moderately ramified actions in positive characteristic.https://www.zbmath.org/1456.140062021-04-16T16:22:00+00:00"Lorenzini, Dino"https://www.zbmath.org/authors/?q=ai:lorenzini.dino-j"Schröer, Stefan"https://www.zbmath.org/authors/?q=ai:schroer.stefanSummary: In characteristic 2 and dimension 2, wild \(\mathbb{Z}/2\mathbb{Z} \)-actions on \(k[[u, v]]\) ramified precisely at the origin were classified by \textit{M. Artin} [Proc. Am. Math. Soc. 52, 60--64 (1975; Zbl 0315.14015)], who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic \(p>0\) and dimension \(n\ge 2\) arising from certain non-linear actions of \(\mathbb{Z}/p\mathbb{Z}\) on the formal power series ring \(k[[u_1,\dots,u_n]]\). These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when \(p=2\). In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.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)