Toric sheaves and flips

TL;DR

This paper studies how toric flips affect the stability of equivariant sheaves, providing criteria for stability preservation and analyzing the behavior of polystability across certain subcategories.

Contribution

It introduces a numerical criterion for slope stability preservation and characterizes polystability behavior under toric flips for specific sheaf subcategories.

Findings

Numerical criterion for slope stability preservation.

Characterization of polystability preservation across subcategories.

Analysis of stability behavior under toric flips.

Abstract

Any toric flip naturally induces an equivalence between the associated categories of equivariant reflexive sheaves, and we investigate how slope stability behaves through this functor. On one hand, for a fixed toric sheaf, and natural polarisations that make the exceptional loci small, we provide a simple numerical criterion that characterizes when slope stability is preserved through the flip. On the other hand, for a given flip, we introduce full subcategories of logarithmic toric sheaves and characterize when polystability is preserved for all toric sheaves in those subcategories at once.