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Moduli spaces of framed sheaves on compactified Kleinian singularities

Søren Gammelgaard

Abstract

Consider a Kleinian singularity $ \mathbb{C}^2/Γ$, where $ Γ$ is a finite subgroup of $ SL_2(\mathbb{C}) $. In this paper, we construct moduli spaces of framed sheaves on a projective Deligne-Mumford stack compactifying the singularity, and we show that these moduli spaces are quasiprojective schemes.

Moduli spaces of framed sheaves on compactified Kleinian singularities

Abstract

Consider a Kleinian singularity , where is a finite subgroup of . In this paper, we construct moduli spaces of framed sheaves on a projective Deligne-Mumford stack compactifying the singularity, and we show that these moduli spaces are quasiprojective schemes.
Paper Structure (10 sections, 15 theorems, 46 equations)

This paper contains 10 sections, 15 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coherent sheaves on projective Deligne-Mumford stacks
  4. Properties of coherent sheaves on a projective Deligne-Mumford stack
  5. Moduli spaces of framed sheaves on projective stacks
  6. Compactified Kleinian singularities and sets of framed sheaves
  7. A scheme structure for $Y_{\mathbf r,n}$
  8. Hilbert polynomials of framed sheaves on $\mathcal{X}$
  9. Boundedness and regularity
  10. Stability of framed sheaves on $\mathcal{X}$

Key Result

theorem 1.1

Choose a nonnegative integer $n$ and a finite-dimensional $\Gamma$-representation $V$, and let $\mathscr{R}$ be the sheaf on $D$ such that $(\pi|_D)^* \mathscr{R} =(\mathcal{O}_L\otimes V)$. There is a fine moduli space $\mathbf{Y}_{\mathbf r,n}$ parametrising isomorphism classes of pairs $(\mathscr

Theorems & Definitions (44)

  • theorem 1.1
  • definition 2.1: Kresch
  • remark 2.2
  • definition 2.3: BruzzoSala
  • definition 2.4: Nironi
  • remark 2.5
  • definition 2.6
  • definition 2.7: BruzzoSala
  • lemma 2.8
  • proof
  • ...and 34 more