Mukai-Fourier transform

The Mukai-Fourier transform is a transformation used in algebraic geometry. It is somewhat analogous to the classical Fourier transform used in analysis.

Definition

Let $X$ be an abelian variety and $hat\; X$ be its dual variety. We denote by $mathcal\; P$ the Poincaré bundle on

:$X\; imes\; hat\; X,$

normalized to be trivial on the fibers at zero. Let $p$ and $hat\; p$ be the canonical projections.

The Fourier-Mukai functor is then:$Rmathcal\; S:\; mathcal\; F\; in\; D(X)\; mapsto\; Rhat\; p\_ast\; (p^ast\; mathcal\; F\; otimes\; mathcal\; P)\; in\; D(hat\; X)$

The notation here: "D" means derived category of coherent sheaves, and "R" is the higher direct image functor, at the derived category level.

There is a similar functor

:$Rwidehat\{mathcal\; S\}\; :\; D(hat\; X)\; o\; D(X)$.

Properties

Let $g$ denote the dimension of $X$.

The Fourier-Mukai transformation is nearly involutive ::$Rmathcal\; S\; circ\; Rwidehat\{mathcal\; S\}\; =\; (-1)^ast\; [-g]$

It transforms Pontrjagin product in tensor product and conversely.:$Rmathcal\; S(mathcal\; F\; ast\; mathcal\; G)\; =\; Rmathcal\; S(mathcal\; F)\; otimes\; Rmathcal\; S(mathcal\; G)$:$Rmathcal\; S(mathcal\; F\; otimes\; mathcal\; G)\; =\; Rmathcal\; S(mathcal\; F)\; ast\; Rmathcal\; S(mathcal\; G)\; [g]$

References

*cite journal
last=Mukai
first=Shigeru
authorlink=Shigeru Mukai
title=Duality between $D(X)$ and $D(hat\; X)$ with its application to Picard sheaves
journal=Nagoya Mathematical Journal
volume=81
date=1981
pages=153–175
id=ISSN 0027-7630
url=http://projecteuclid.org/euclid.nmj/1118786312