# The Fourier-Mukai transform made easy

## TL;DR

This paper introduces a simplified approach to the Fourier-Mukai transform on abelian varieties, facilitating easier understanding and proofs of key theorems related to sheaf properties and global generation.

## Contribution

It presents a modified definition of the Fourier-Mukai transform that simplifies formula derivation and provides concise proofs for important theorems in algebraic geometry.

## Key findings

- Simplified Fourier-Mukai transform formula
- Short proof of GV-sheaves characterization
- Proof that M-regularity implies global generation

## Abstract

We propose a slightly modified definition for the Fourier-Mukai transform (on abelian varieties) that makes it much easier to remember various formulas. As an application, we give relatively short proofs for two important theorems: the characterization of GV-sheaves in terms of vanishing, due to Hacon; and fact that M-regularity implies (continuous) global generation, due to Pareschi and Popa.