Fourier–Mukai transform

The Fourier–Mukai transform or 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 \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 transforms Pontrjagin product in tensor product and conversely.

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