Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities

TL;DR

This paper investigates maximal Cohen-Macaulay sheaves on symplectic singularities, using resolutions and duality to construct and characterize such sheaves, with explicit examples on nilpotent matrix varieties.

Contribution

It introduces a method to lift Cohen-Macaulay sheaves to resolutions and characterizes when their pushforwards remain Cohen-Macaulay, providing explicit constructions on specific symplectic singularities.

Findings

Constructed many indecomposable Cohen-Macaulay sheaves on nilpotent matrix varieties.

Characterized reflexive sheaves on cotangent bundles with Cohen-Macaulay pushforwards.

Extended constructions to higher-dimensional cases, generalizing previous results.

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 $T^*\mathbb{P}^n \to \mathcal{N}_{n+1,1}$.