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Moduli of sheaves on hyperkähler manifolds

Kieran G. O'Grady

Abstract

We survey recent advances in the theory of moduli spaces of stable sheaves on hyperkähler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the techniques which can be extended to higher dimensions.

Moduli of sheaves on hyperkähler manifolds

Abstract

We survey recent advances in the theory of moduli spaces of stable sheaves on hyperkähler manifolds of dimension greater than . We start by recalling the well-known theory in dimension , i.e.~for surfaces, emphasizing the techniques which can be extended to higher dimensions.
Paper Structure (48 sections, 40 theorems, 100 equations)

This paper contains 48 sections, 40 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Background and goal of the paper
  3. Outline of the paper
  4. Conventions and notation
  5. Moduli of semistable sheaves on $K3$ surfaces
  6. Deformations of sheaves
  7. Stability and semistability
  8. Gieseker-Maruyama (semi)stability, and moduli spaces
  9. Slope (semi)stability
  10. Variation of slope (semi)stability
  11. Motivation and overview
  12. Slope with respect to nef classes
  13. Suitability
  14. Wall-and-chamber decomposition of the ample cone
  15. Stability for surfaces fibered over a curve
  16. ...and 33 more sections

Key Result

Theorem 2.1.1

Let $S$ be a projective $K3$ surface, and let $\mathscr{F}$ be a simple torsion-free sheaf on $S$. Then the deformation space $\mathop{\mathrm{Def}}\nolimits(\mathscr{F})$ is smooth, and its dimension is given by

Theorems & Definitions (113)

  • Theorem 2.1.1: Mukai-Artamkin
  • proof
  • Theorem 2.1.2: huang:modcoppieiacono-manetti:defcoppie
  • Corollary 2.1.3
  • proof
  • Theorem 2.1.4: Horikawa
  • proof
  • Corollary 2.1.5
  • Definition 2.2.1
  • Theorem 2.2.2: Gieseker gieseker:modfasci - Maruyama maruyama:modfasci
  • ...and 103 more