A note on singular moduli spaces of sheaves on K3 surfaces

This paper studies deformations and birational maps between singular moduli spaces of torsion free semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that when the greatest common divisor of the rank and the first Chern class is 2, two such moduli spaces of the same dimension can be connected by deformations and birational maps.     

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     

Symplectic structures on moduli spaces of framed sheaves on surfaces

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.     

Poisson structures on moduli spaces of sheaves over Poisson surfaces

Summary We introduce and study the notion of Poisson surface. We prove that the choice of a Poisson structure on a surfaceS canonically determines a Poisson structure on the moduli space of stable sheaves onS. This result generalizes previous results obtained by Mukai [14], for abelian orK3 surfaces, and by Tyurin [16].Oblatum 13-VI-1994 & 22-III-1995This article was processed by the author using thepjourlm style file from Springer-Verlag     

Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure.     

On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Fix a smooth very ample curve C on a K3 or abelian surface X. Let $ \mathcal{M} $ denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on $ \mathcal{M} $. On the other hand, $ \mathcal{M} $ has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on $ \mathcal{M} $ coincides with the pullback of the symplectic form on the Hilbert scheme.     

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.     

Rationality of moduli spaces of torsion free sheaves over rational surfaces

In this paper we show that for rational ruled surfaces many moduli spaces of torsion free sheaves with given Chern classes are rational. We deal with the case that the first Chern classc ₁ satisfiesc ₁.F=0 for a fibreF of the ruling. The main tool are priority sheaves introduced by Hirschowitz-Laszlo and Walter, which enable us to reduce the problem to the construction of a family of sheaves over a big enough rational base.     

Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups

In a series of our papers with Carlo Madonna (2002–2008), we described self-correspondences of a K3 surface over ℂ via moduli of sheaves with primitive isotropic Mukai vectors for the Picard number 1 or 2 of the K3 surfaces. Here we give a natural and functorial answer to the same problem for an arbitrary Picard number. As an application, we characterize, in terms of self-correspondences via moduli of sheaves, K3 surfaces with reflective Picard lattice, that is, when the automorphism group of the lattice is generated by reflections up to finite index. It is known since 1981 that the number of reflective hyperbolic lattices is finite. We also formulate some natural unsolved related problems.     

Semistable sheaves on homogeneous spaces and abelian varieties

In this paper we prove that semistable sheaves with zero Chern classes on homogeneous spaces are trivial and semistable sheaves on abelian varieties with zero Chern classes are filtered by line bundles numerically equivalent to zero. The method consists in reducing modp and then showing that the Frobenius morphism preserves semistability on the above class of varieties. For technical reasons, we have to assume boundedness of semistable sheaves in charp.     

Birational properties of moduli spaces of stable locally free rank-2 sheaves on algebraic surfaces

LetX be a smooth algebraic surface over the complex number field. Fix a polarizationL, an invertible sheafc ₁ and an integerc ₂ such that (4c ₂-c ₁ ² ) is positive. letM _L(c ₁,c ₂) be the moduli space ofL-stable locally free rank-2 sheaves onX with chern classesc ₁ andc ₂ respectively, and let ξ be a numerical equivalence class defining a nonempty wall of type (c ₁,c ₂). We study the properties ofE _ξ(c ₁,c ₂) and obtain estimations for its dimension. Then, we discuss the existence of trivial polarizations, and determine the birational structures of moduli spacesM _L(c ₁,c ₂) whenX is a minimal surface of general type and (4c ₂-c ₁ ² ) is sufficiently large.     

Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

Let M be a smooth and compact moduli space of stable coherent sheaves on a projective surface S with an effective (or trivial) anti-canonical line bundle. We find generators for the cohomology ring of M, with integral coefficients. When S is simply connected and a universal sheaf E exists over S×M, then its class [E] admits a Künneth decomposition as a class in the tensor product of the topological K-rings. The generators are the Chern classes of the Künneth factors of [E] in . The general case is similar.     

Hecke correspondences for smooth moduli spaces of sheaves

Publications mathématiques de l'IHÉS - We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and...