Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities

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 $T^{\ast }{\mathbb{P}}^2\to {\mathcal{N}}_{3,1}$, 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 $T^{\ast }{\mathbb{P}}^2$ 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 $T^{\ast }{\mathbb{P}}^n\to {\mathcal{N}}_{n+1,1}$.