Prüfer Sheaves and Generic Sheaves Over the Weighted Projective Lines and Elliptic Curves

In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.     

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.     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

Das Spektrum torsionsfreier Garben I

In this paper we study torsion free sheaves of arbitrary rank on protective spaces. These sheaves naturally occur in the closure of the moduli spaces of stable vector bundles. We generalize some of the techniques and results of Hartshorne [3], [4] to torsion free sheaves. Applications will be given in another paper.     

A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )$. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of ${\mathbb T}$ and show that no smooth genus 5 curve on such a surface can contain a $g^1_3$. We also describe explicitly the singular divisors in the linear system |2?|.     

On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.      

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.     

Lazarsfeld-Mukai reflexive sheaves and their stability

Consider an ample and globally generated line bundle L on a smooth projective variety X of dimension N≥2 over ?. Let D be a smooth divisor in the complete linear system of L. We construct reflexive sheaves on X by an elementary transformation of a trivial bundle on X along certain globally generated torsion-free sheaves on D. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the μ_L-(semi)stability of such reflexive sheaves under certain conditions.     

Classification of semistable sheaves on a rational curve with one node

We classify semistable sheaves on a rational curve with one node. The results are based on the classification of indecomposable torsion-free sheaves due to Drozd and Greuel (2001) [3], where the sheaves are described in terms of certain combinatorial data. We translate the condition of semistability into this combinatorial language and solve the so obtained problem.     

The elementary transformation of vector bundles on regular schemes

We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.

     

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

We extend Nadel’s results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal subvarieties on toric del Pezzo surfaces which do not admit Kähler–Einstein metrics. We also show that one can define multiplier ideal sheaves for Kähler–Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.     

Formal and Tempered Solutions of Regular D-Modules

A new family of sheaves has been recently studied by M. Kashiwara and P. Schapira generalizing to constructible sheaves the notion of moderate and formal cohomology. We prove comparison theorems when we regard these sheaves as solutions of a D-module. These results are natural generalizations of those of Y. Laurent and the author.     

Perverse <Emphasis Type="Italic">t</Emphasis>-structure on Artin stacks

In this paper we develop the theory of perverse sheaves on Artin stacks continuing our earlier study of lisse-étale sheaves on stacks in Laszlo and Olsson (The six operations for sheaves on Artin stacks I: Finite Coefficients. Publ Math IHéS, 2008; The six operations for sheaves on Artin stacks II: Adic Coefficients. Publ Math IHéS, 2008).     

On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics

We decompose each moduli space of semi-stable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a non-reductive group. We find locally free resolutions of length one for all these sheaves and describe them.     

Stability and Fourier–Mukai transforms on elliptic fibrations

We systematically develop Bridgeland's [7] and Bridgeland–Maciocia's [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1–1 correspondences between fibrewise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.     

Some Examples of Tilt-Stable Objects on Threefolds

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarization ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.