A note on stable toric sheaves of low rank

TL;DR

This paper constructs explicit examples of stable equivariant reflexive sheaves of low rank on polarized toric varieties, contrasting with known splitting results for vector bundles of lower rank, and analyzes their singular loci.

Contribution

It provides the first explicit examples of stable equivariant reflexive sheaves of certain low ranks on arbitrary polarized toric varieties, expanding understanding beyond splitting theorems.

Findings

Explicit stable reflexive sheaves exist for ranks up to the dimension plus Picard rank.

The dimension of the singular locus of these sheaves is strictly less than the ambient dimension minus the rank.

Contrasts with known splitting results for vector bundles of lower rank.

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 $n - r$.