Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915855093202944 |
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| author | Xu, Shang |
| author_facet | Xu, Shang |
| contents | 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 $T^*\mathbb{P}^n \to \mathcal{N}_{n+1,1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11227 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities Xu, Shang Algebraic Geometry 14B05 (Primary) 14E15, 14F06, 13C14 (Secondary) 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 $T^*\mathbb{P}^n \to \mathcal{N}_{n+1,1}$. |
| title | Constructing Maximal Cohen-Macaulay Sheaves on Symplectic Singularities |
| topic | Algebraic Geometry 14B05 (Primary) 14E15, 14F06, 13C14 (Secondary) |
| url | https://arxiv.org/abs/2603.11227 |