Mukai-Fourier transform
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The Mukai-Fourier transform is a transformation used in algebraic geometry. It is somewhat analogous to the classical Fourier transform used in analysis.
[edit] Definition
Let X be an abelian variety and 
be its dual variety. We denote by 
the Poincaré bundle on
normalized to be trivial on the fibers at zero. Let p and 
be the canonical projections.
The Fourier-Mukai functor is then
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
.
[edit] 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]](../../../math/a/c/6/ac6303137b90dc29b18ff292c0709722.png)
It transforms Pontrjagin product in tensor product and conversely.
![R\mathcal S(\mathcal F \otimes \mathcal G) = R\mathcal S(\mathcal F) \ast R\mathcal S(\mathcal G)[g]](../../../math/a/e/2/ae282b672bca8a904e2ab319a95eddea.png)
[edit] References
- Mukai, Shigeru (1981). "Duality between D(X) and
with its application to Picard sheaves". Nagoya Mathematical Journal 81: 153-175. ISSN 0027-7630.
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