Semistable abelian varieties with good reduction outside 15

We show that there are no non-zero semi-stable abelian varieties over ${{\bf Q}(\sqrt{5})}$ with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X ₀(15).     

Moduli spaces of stable sheaves on abelian surfaces

In this paper, we consider basic problems on moduli spaces of stable sheaves on abelian surfaces. Our main assumption is the primitivity of the associated Mukai vector. We determine the deformation types, albanese maps, Bogomolov factors and their weight 2 Hodge structures. We also discuss the deformation types of moduli spaces of stable sheaves on K3 surfaces. Received: 28 February 2000 / Revised version: 15 September 2000 / Published online: 24 September 2001     

Vanishing theorems for constructible sheaves on abelian varieties over finite fields

Let \(\kappa \) be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing \(\kappa \). For a perverse \(\overline{\mathbb {Q}}_\ell \)-adic sheaf \(K_0\) on an abelian variety \(X_0\) over \(\kappa \), let K and X denote the base field extensions of \(K_0\) and \(X_0\) to k. Then, the aim of this note is to show that the Euler–Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. \(\chi (X,K)=\sum _\nu (-1)^\nu \dim _{\overline{\mathbb {Q}}_\ell }(H^\nu (X,K))\ge 0\). This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that \(\chi (X,K)=0\) implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.     

On moduli spaces of sheaves on K3 or abelian surfaces

We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the existence of new examples of projective irreducible symplectic manifolds lying birationally over components of the moduli spaces of one-dimensional semistable sheaves on K3 surfaces, and over components of many of the moduli spaces of two-dimensional sheaves on K3 surfaces, in particular, of those for rank two sheaves.     

Foliations in moduli spaces of abelian varieties

We study moduli spaces of polarized abelian varieties in positive characteristic. Our final goal will be to understand Hecke orbits in such spaces. This paper provides one of the tools. For a given -divisible group, all abelian varieties which give rise to this group have moduli points in a locally closed subset of the moduli space; we call an irreducible component of this subset a central leaf. Newton polygon strata are foliated by such leaves. Moreover, iterated -isogenies give a second leaf structure, which was already known under the name of Rapoport-Zink spaces. Any Newton polygon stratum is, up to a finite morphism, isomorphic to a product of an isogeny leaf and a finite cover of a central leaf. We conjecture that any Hecke--orbit is dense in the corresponding central leaf.

     

On the derived categories of coherent sheaves on some homogeneous spaces

To A. Grothendieck on his 60-th birthday     

Semistable laws on topological vector spaces

Summary In this paper we discuss three types of results: Firstly, we present two Lévy-Hinin type representations of Poisson type infinitely divisible (i.d.) laws on certain topological vector (TV) spaces; one of these complements a known representation due to Dettweiler. Secondly, we define and characterize r-semistable laws on locally convex TV spaces and also obtain good representation of their characteristic functions. Finally, we discuss the existence and the continuity of the semigroup { _tt>0} of i.d. laws on locally convex TV spaces. These complement similar known results of Siebert.The research of this author is partially supported by the Office of Naval Research under contract No. N0014-78-C-0468     

On the embedding of abelian varieties in projective spaces

Summary In this note it is proved that forα ⩾3 an abelian variety of dimension d cannot be embedded in a projective space of dimenrion 2d. Dedicated to ProfessorBeniamino Segre on the occasion of his 70th birthday Entrata in Redazione il 7 giugno 1973.     

Equivariant sheaves on flag varieties

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.     

Tilting sheaves on toric varieties

In [19], A. King states the following conjecture: Any smooth complete toric variety has a tilting bundle whose summands are line bundles. The goal of this paper is to prove Kings conjecture for the following types of smooth complete toric varieties: (i) Any d-dimensional smooth complete toric variety with splitting fan. (ii) Any d-dimensional smooth complete toric variety with Picard number 2. (iii) The blow up of any smooth complete minimal toric surface at T-invariants points.Mathematics Subject Classification (1991): 14F05; 14M25Partially supported by BFM2001-3584.     

On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Let J be an abelian surface with a generic ample line bundle . For n≥1, the moduli space M_J(2,0,2n) of (1)-semistable sheaves F of rank 2 with Chern classes c₁(F)=0, c₂(F)=2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of M_J(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of M_J(2,0,2n) for n≥2. Jaeyoo Choy was partially supported by KRF 2003-070-C00001 Young-Hoon Kiem was partially supported by a KOSEF grant R01-2003-000-11634-0.     

Coherent sheaves on quiver varieties and categorification

We construct geometric categorical $\mathfrak g $ actions on the derived category of coherent sheaves on Nakajima quiver varieties. These actions categorify Nakajima’s construction of Kac–Moody algebra representations on the K-theory of quiver varieties. We define an induced affine braid group action on these derived categories.     

Character sheaves on certain spherical varieties

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.