Toric sheaves, stability and fibrations

TL;DR

This paper investigates the stability of equivariant reflexive sheaves on toric varieties under toric fibrations, providing criteria for stability preservation and applications to tangent sheaves and moduli spaces.

Contribution

It establishes conditions under which slope stability is preserved under pullbacks along toric fibrations, including blow-ups and trivial fibrations, and explores implications for moduli spaces.

Findings

Stability is preserved under pullbacks with adiabatic polarisations.

Necessary and sufficient conditions for stability in strictly semistable cases.

Construction of stable perturbations of tangent sheaves through polarisation changes or blow-ups.

Abstract

For an equivariant reflexive sheaf over a polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic polarisations. In the strictly semistable situation, under locally freeness assumptions, we provide a necessary and sufficient condition on the graded object to ensure stability of the pulled back sheaf. As applications, we provide various stable perturbations of semistable tangent sheaves, either by changing the polarisation, or by blowing-up a subvariety. Finally, our results apply uniformly in specific flat families and induce injective maps between the associated moduli spaces.