A note on stable toric sheaves of low rank

Carl Tipler

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arXiv · 2307.02822

A note on stable toric sheaves of low rank

Carl Tipler

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Abstract

Kaneyama and Klyachko have shown that any torus equivariant vector bundle of rank $r$ over $\mathbb{CP}^n$ splits if $r < n$. In particular, any such bundle is not slope stable. In contrast, we provide explicit examples of stable equivariant reflexive sheaves of rank $r$ on any polarised toric variety $(X, L)$, for $2 \leq r < \mathrm{dim}(X) + \mathrm{rank}(\mathrm{Pic}(X))$, and show that the dimension of their singular locus is strictly bounded by $n - r$.

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Carl Tipler. 2023-07-06. A note on stable toric sheaves of low rank. https://arxiv.org/abs/2307.02822

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