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A note on stable toric sheaves of low rank

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arxiv 2307.02822 v1 pith:BDQYPEHO submitted 2023-07-06 math.AG

classification math.AG
keywords rankmathrmstablebundleequivariantsheavestoricbounded
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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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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups

    math.AG 2025-06 accept novelty 6.0 of 10

    P-critical connections generalize Z-critical connections; on toric varieties P-positivity is checked finitely, and uniform P-positivity survives point blow-ups.

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