Fourier-Mukai transforms and the wall-crossing behavior for Bridgeland's stability conditions

TL;DR

This paper explores how Fourier-Mukai transforms affect Bridgeland stability conditions, relating them to Gieseker stability, and demonstrates the birational equivalence of moduli spaces of stable sheaves on abelian surfaces through wall-crossing analysis.

Contribution

It clarifies the relation between Fourier-Mukai transforms and stability conditions, and proves birational equivalence of moduli spaces via wall-crossing techniques.

Findings

Fourier-Mukai transforms preserve Bridgeland stability conditions.

Moduli spaces of stable sheaves are birationally equivalent under certain isometries.

Wall-crossing behavior explains the birational relationships.

Abstract

Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. We explain the relation with Gieseker stability. By studying the wall-crossing behavior, we reprove that the moduli spaces of stable sheaves on abelian surfaces are birationally equivalent, if the associated Mukai vectors are related by isometries of the Mukai lattice.