On secant dimensions and identifiability of flag varieties

We investigate the secant dimensions and the identifiability of flag varieties parametrizing flags of subspaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.     

Veronese Varieties Over Finite Fields and Their Projections

Inthis paper Veronese varieties of degree d over aGalois field are studied. We also show that some of known capsembedded into classical varieties always are projections of Veronesevarieties.     

Holomorphic symmetric differentials and a birational characterization of abelian varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.     

On Identity Bases of Exclusion Varieties for Monoids

An exclusion variety is a variety that is maximal with respect to not containing some semigroup. It is shown that if the bases of all exclusion varieties for some periodic semigroup S are known, then the bases of exclusion varieties for the monoid S ¹ can be computed.     

Parameters of the codes over affine varieties

We derive a lower bound of the generalized Hamming weights of the codes over affine varieties, which are defined by appropriate sequences of rational polynomials over varieties.     

Derived categories and Kummer varieties

We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.      

Cohomologie des variétés de Deligne-Lusztig

We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element w₀ (some indeterminacies remain for G₂). We use this to construct Hecke algebra actions on the cohomology of varieties associated to w₀ or its square, for groups of arbitrary rank. In the subsequent work [F. Digne, J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya J. Math. 183 (2006)], we construct actions associated to more general regular elements and we study their traces on cohomology.     

Sanya Lectures: Geometry of Spherical Varieties

These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.     

Complete Congruences on the Lattice of Rees–Sushkevich Varieties

We study the lattice ?(RS_n) of subvarieties of the variety of semigroups generated by completely 0-simple semigroups over groups with exponent dividing n, with a particular focus on the lattice ??(RS_n) consisting of those varieties that are generated by completely 0-simple semigroups. The sublattice of ??(RS_n) consisting of the aperiodic varieties is described and several endomorphisms of ?(RS_n) considered. The complete congruence on ??(RS_n) that relates varieties containing the same aperiodic completely 0-simple semigroups is considered in some detail.     

Identifying quadric bundle structures on complex projective varieties

In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.      

Algebraic independence in the Grothendieck ring of varieties

We give sufficient cohomological criteria for the classes of given varieties over a field to be algebraically independent in the Grothendieck ring of varieties over and construct some examples.

     

Support Varieties for Selfinjective Algebras

Support varieties for any finite dimensional algebra over a field were introduced in (Proc. London Math. Soc. 88 (3) (2004) 705–732) using graded subalgebras of the Hochschild cohomology ring. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, the variety of periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true. As a corollary of a more general result we show that Webbs theorem generalizes to finite dimensional cocommutative Hopf algebras.Received November 2003Mathematics Subject Classifications (2000) Primary: 16E40, 16G10, 16P10, 16P20; Secondary: 16G70.     

On complete intersections with trivial canonical class

We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev–Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow.     

高维代数簇计算的一些结果

本文在［１］的基础上给出计算高维代数簇的特征值方法,进一步结果,得到了不可约集的母点和对应于极大无关变元集的多项式方程组的孤立解之间对应关系的进一步刻画．     

Rational Hypergeometric Functions

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.     

A note on the Chow groups of projective determinantal varieties; an appendix to the paper by Grzegorz Kapustka and Michal Kapustka on “A cascade of determinantal Calabi‐Yau threefolds”

We investigate the Chow groups of projective determinantal varieties and those of their strata of matrices of fixed rank, using Chern class computations (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Families of Algebraic Varieties Parametrized by Topological Spaces

We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between the category of bivariant coherent sheaves and the category of sheaves of finitely generated modules.     

Quiver varieties and Lusztig's algebra

We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.     

On the characterization of complex Shimura varieties

In this paper we recall basic properties of complex Shimura varieties and show that they actually characterize them. This characterization immediately implies the explicit form of Kazhdan's theorem on the conjugation of Shimura varieties. It also implies the existence of unique equivariant models over the reflex field of Shimura varieties corresponding to adjoint groups and the existence of a p-adic uniformization of certain unitary Shimura varieties. In the appendix we give a modern formulation and a proof of Weil's descent theorem.