Stability conditions on free abelian quotients

TL;DR

This paper investigates stability conditions on varieties that are quotients of smooth projective varieties by finite abelian groups, establishing isomorphisms between invariant stability conditions on covers and quotients, and exploring implications for varieties with non-finite Albanese morphisms.

Contribution

It provides an analytic isomorphism between G-invariant stability conditions on covers and residual-invariant conditions on quotients, and describes stability manifolds for certain abelian quotients.

Findings

Established an isomorphism between G-invariant and residual-invariant stability conditions.

Described connected components of stability manifolds for varieties with finite Albanese morphism.

Counterexamples to a conjecture on the Le Potier function were provided.

Abstract

We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is an analytic isomorphism between $G$-invariant geometric stability conditions on the cover and geometric stability conditions on the quotient that are invariant under the residual action of the group $\widehat{G}$ of irreducible representations of $G$. We apply our results to describe a connected component inside the stability manifolds of free abelian quotients when the cover has finite Albanese morphism. This applies to varieties with non-finite Albanese morphism which are free abelian quotients of varieties with finite Albanese morphism, such as Beauville-type and bielliptic surfaces. This gives a partial answer to a question raised by Lie Fu, Chunyi Li, and Xiaolei Zhao: if a variety $X$ has non-finite Albanese morphism, does there always exist a non-geometric stability condition on $X$? We also give counterexamples to a conjecture of Fu--Li--Zhao concerning the Le Potier function, which characterises Chern classes of slope-semistable sheaves. As a result of independent interest, we give a description of the set of geometric stability conditions on an arbitrary surface in terms of a refinement of the Le Potier function. This generalises a result of Fu--Li--Zhao from Picard rank $1$ to arbitrary Picard rank.