In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y).
Most natural functors, including basic ones like pushforwards and pullbacks, are of this type.
These kinds of functors were introduced by Mukai (1981) in order to prove an equivalence between the derived categories of coherent sheaves on an abelian variety and its dual.
That equivalence is analogous to the classical Fourier transform that gives an isomorphism between tempered distributions on a finite-dimensional real vector space and its dual.
Let X and Y be smooth projective varieties, K ∈ Db(X×Y) an object in the derived category of coherent sheaves on their product.
, taken as a kernel, produces the identity functor on Db(X).
For a morphism f:X→Y, the structure sheaf of the graph Γf produces a pushforward when viewed as an object in Db(X×Y), or a pullback when viewed as an object in Db(Y×X).
is then There is a similar functor If the canonical class of a variety is ample or anti-ample, then the derived category of coherent sheaves determines the variety.
[1] In general, an abelian variety is not isomorphic to its dual, so this Fourier–Mukai transform gives examples of different varieties (with trivial canonical bundles) that have equivalent derived categories.
Deninger & Murre (1991) have used the Fourier-Mukai transform to prove the Künneth decomposition for the Chow motives of abelian varieties.