Fourier–Mukai transform

In algebraic geometry, the Fourier–Mukai transform or Mukai–Fourier transform, introduced by Mukai (1981), is an isomorphism between the derived categories of coherent sheaves on an abelian variety and its dual. It is analogous to the classical Fourier transform that gives an isomorphism between tempered distributions on a finite-dimensional real vector space and its dual.

If the canonical class of a variety is positive or negative, then the derived category of coherent sheaves determines the variety. The Fourier–Mukai transform gives examples of different varieties (with trivial canonical bundle) that have isomorphic derived categories, as in general an abelian variety of dimension greater than 1 is not isomorphic to its dual.

Definition

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

$X\times {\hat X},$

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

The Fourier–Mukai functor is then

$R{\mathcal S}:{\mathcal F}\in D(X)\mapsto R{\hat p}_{\ast }(p^{\ast }{\mathcal F}\otimes {\mathcal P})\in D({\hat X})$

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

$R\widehat {{\mathcal S}}:D({\hat X})\to D(X).\,$

Properties

Let g denote the dimension of X.

The Fourier–Mukai transformation is nearly involutive :

$R{\mathcal S}\circ R\widehat {{\mathcal S}}=(-1)^{\ast }[-g]$

It interchanges Pontrjagin product and tensor product.

$R{\mathcal S}({\mathcal F}\ast {\mathcal G})=R{\mathcal S}({\mathcal F})\otimes R{\mathcal S}({\mathcal G})$

$R{\mathcal S}({\mathcal F}\otimes {\mathcal G})=R{\mathcal S}({\mathcal F})\ast R{\mathcal S}({\mathcal G})[g]$

References

• Huybrechts, D. (2006), Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, doi:10.1093/acprof:oso/9780199296866.001.0001, ISBN 978-0-19-929686-6, MR 2244106, ISBN 978-0-19-929686-6
• Mukai, Shigeru (1981). "Duality between $D(X)$ and $D({\hat X})$ with its application to Picard sheaves". Nagoya Mathematical Journal. 81: 153–175. ISSN 0027-7630.