A note on stable toric sheaves of low 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 $2\le r<\dim (X)+\mathrm{rank}(\mathrm{Pic}(X))$, and show that the dimension of their singular locus is strictly bounded by $n-r$.

7 pages