Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 4

Special issue celebrating the work of Herb Clemens

Guest Editor: Ron Donagi

The Fourier–Mukai transform made easy

Pages: 1749 – 1770

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a14


Christian Schnell (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)


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 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.


Fourier–Mukai transform, abelian variety, GV-sheaf, $m$-regularity

The author was partially supported by grant DMS-1404947 from the National Science Foundation, and by a Centennial Fellowship from the American Mathematical Society.

Received 14 December 2021

Received revised 31 December 2021

Accepted 3 January 2022

Published 25 October 2022