2023/07/03 by Hannah Dell · 3 citations
Mathematics · #math.AG
paper · pdf · doi:10.46298/epiga.2025.11719
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 \widehatG 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.