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Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities

Shang Xu

Abstract

In this paper, we study maximal Cohen-Macaulay sheaves on symplectic singularities. These sheaves generate the singularity categories and thus measure how far a singularity is from being smooth. We lift maximal Cohen-Macaulay sheaves on a singular variety to reflexive sheaves on its resolution and use Grothendieck duality to study their cohomological vanishing. We work this out in detail for the resolution $T^*\mathbb{P}^2 \rightarrow \mathcal{N}_{3,1}$, where $\mathcal{N}_{j,k}$ denotes the variety of nilpotent $j\times j$ matrices of rank at most $k$. In this case, we characterize the reflexive sheaves on $T^*\mathbb{P}^2$ whose pushforwards are maximal Cohen-Macaulay, and use vanishing results on $\mathbb{P}^2$ to construct many indecomposable maximal Cohen-Macaulay sheaves on $\mathcal{N}_{3,1}$. We also extend this construction to the resolution $T^*\mathbb{P}^n \to \mathcal{N}_{n+1,1}$.

Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities

Abstract

In this paper, we study maximal Cohen-Macaulay sheaves on symplectic singularities. These sheaves generate the singularity categories and thus measure how far a singularity is from being smooth. We lift maximal Cohen-Macaulay sheaves on a singular variety to reflexive sheaves on its resolution and use Grothendieck duality to study their cohomological vanishing. We work this out in detail for the resolution , where denotes the variety of nilpotent matrices of rank at most . In this case, we characterize the reflexive sheaves on whose pushforwards are maximal Cohen-Macaulay, and use vanishing results on to construct many indecomposable maximal Cohen-Macaulay sheaves on . We also extend this construction to the resolution .
Paper Structure (13 sections, 51 theorems, 72 equations)

This paper contains 13 sections, 51 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Conventions.
  3. Acknowledgments.
  4. Symplectic Singularities, Nakajima Quiver varieties
  5. Symplectic singularities
  6. Nakajima Quiver Varieties
  7. Maximal Cohen-Macaulay modules on Symplectic singularities
  8. Constructions on $T^*\mathbb{P}^2\rightarrow \mathcal{N}_{3,1}$
  9. Characterization of maximal Cohen-Macaulay condition
  10. Bundles on $\mathbb{P}^2$ pushing forward to maximal Cohen-Macaulay sheaves
  11. Classification of 2-bundles with cohomology vanishing
  12. Construction of higher rank maximal Cohen-Macaulay sheaves
  13. Constructions on $T^*\mathbb{P}^n\rightarrow \mathcal{N}_{n+1,1}$

Key Result

Proposition 1.1

Suppose $\mathcal{F}$ is a reflexive sheaf on $T^*\mathbb{P}^2$. Then $\pi_*\mathcal{F}$ is reflexive. It is maximal Cohen-Macaulay if and only if

Theorems & Definitions (79)

  • Proposition 1.1: Proposition \ref{['prop4.2']}
  • Theorem 1.2: Corollary \ref{['coro4.19']}
  • Theorem 1.3: Corollary \ref{['coro5.9']}
  • Definition 2.1: Symplectic singularity
  • Lemma 2.2
  • Example 2.1: Kleinian surfaces as symplectic resolutions
  • Example 2.2: Springer resolution as a symplectic resolution
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 69 more