Semi-homogeneous sheaves, Fourier-Mukai transforms and moduli of stable sheaves on abelian surfaces

TL;DR

This paper explores the structure of stable sheaves on abelian surfaces using semi-homogeneous sheaves and Fourier-Mukai transforms, proving a longstanding conjecture and describing moduli space correspondences.

Contribution

It introduces semi-homogeneous presentations and demonstrates how Fourier-Mukai transforms affect stable sheaves, confirming Mukai's conjecture.

Findings

Proof of Mukai's conjecture from the 1980s

Explicit description of birational correspondences

Behavior of stable sheaves under Fourier-Mukai transforms

Abstract

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.