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Moduli spaces of sheaves via affine Grassmannians

Daniel Halpern-Leistner, Andres Fernandez Herrero, Trevor Jones

Abstract

We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two familiar moduli problems: the stack of $Λ$-modules and the stack of pairs. In both examples, we construct a $Θ$-stratification of the stack, defined in terms of a polynomial numerical invariant, and we construct good moduli spaces for the open substacks of semistable points. One of the essential ingredients is the construction of higher dimensional analogues of the affine Grassmannian for the moduli problems considered.

Moduli spaces of sheaves via affine Grassmannians

Abstract

We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two familiar moduli problems: the stack of -modules and the stack of pairs. In both examples, we construct a -stratification of the stack, defined in terms of a polynomial numerical invariant, and we construct good moduli spaces for the open substacks of semistable points. One of the essential ingredients is the construction of higher dimensional analogues of the affine Grassmannian for the moduli problems considered.

Paper Structure

This paper contains 28 sections, 39 theorems, 85 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Infinite dimensional GIT
  3. Comparison with the classical approach
  4. Preliminaries
  5. Notation
  6. Pure sheaves
  7. Hilbert polynomials and Gieseker semistability
  8. Some related moduli problems
  9. Moduli of pairs
  10. Moduli of $\Lambda$-modules
  11. $\Theta$-stratifications and numerical invariants
  12. Rational maps and affine Grassmannians
  13. Regular principal subschemes and twists
  14. Stacks of rational maps
  15. The affine Grassmannian for pure sheaves
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $X$ be a projective scheme of finite presentation over a scheme $S$. To any relatively ample line bundle $\mathcal{O}_X(1)$ on $X$, one can associate a polynomial-valued numerical invariant $\nu$ on $\mathop{\mathrm{Coh}}\nolimits^{d}(X)$ (Definition defn: poly numerical invariant on coh). The r

Figures (15)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 10 more figures

Theorems & Definitions (126)

  • Theorem 1.1: = Theorems \ref{['thm: existence of weak theta stratification for pairs']}, \ref{['thm: moduli space pairs']}
  • Remark 1.2
  • Theorem 1.3: = Theorem \ref{['thm: theta stratification pure sheaves']}
  • Remark 1.4
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Moduli of pure sheaves
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 116 more