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.^{[clarification needed]}

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

• 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. http://projecteuclid.org/euclid.nmj/1118786312.