Stability manifolds of varieties with finite Albanese morphisms

TL;DR

This paper proves that for certain complex varieties with finite Albanese morphisms, all Bridgeland stability conditions are geometric, and describes the connected, contractible stability manifolds for specific irregular surfaces and abelian threefolds.

Contribution

It establishes that all stability conditions are geometric for varieties with finite Albanese morphisms and characterizes the stability manifolds for specific classes of irregular surfaces and abelian threefolds.

Findings

All skyscraper sheaves are stable with the same phase.

Stability manifolds of irregular surfaces and abelian threefolds with Picard rank one are connected.

These stability manifolds are contractible.

Abstract

For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible.