# Fourier-Mukai partners of abelian varieties

## TL;DR

This paper explores the classification of Fourier-Mukai partners of abelian varieties, focusing on semi-homogenous vector bundles and their role in derived equivalences between abelian varieties.

## Contribution

It provides new insights into the conditions under which semi-homogenous vector bundles induce derived equivalences between abelian varieties.

## Key findings

- Characterization of Fourier-Mukai partners via semi-homogenous vector bundles
- Conditions for a semi-homogenous vector bundle to be an image of the structure sheaf
- Analysis of kernels inducing derived equivalences

## Abstract

We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an equivalence is given by a semi-homogenous vector bundle. In particular, given an abelian variety B, and a simple semi-homogenous vector bundle E on B, we will discuss for which E, it can be the image of the structure sheaf of the unit on A under some triangulated equivalence, where A is some abelian variety.