Fourier–Mukai transform

Let X and Y be smooth projective varieties, K ∈ D^b(X×Y) an object in the derived category of coherent sheaves on their product. Denote by q the projection X×Y→X, by p the projection X×Y→Y. Then the Fourier-Mukai transform Φ_K is a functor D^b(X)→D^b(Y) given by

${\displaystyle {\mathcal {F}}\mapsto \mathrm {R} p_{*}\left(q^{*}{\mathcal {F}}\otimes ^{L}K\right)}$

where Rp_* is the derived direct image functor and ${\displaystyle \otimes ^{L}}$ is the derived tensor product.

Fourier-Mukai transforms always have left and right adjoints, both of which are also kernel transformations. Given two kernels K₁ ∈ D^b(X×Y) and K₂ ∈ D^b(Y×Z), the composed functor Φ_K2 ∘ Φ_K1 is also a Fourier-Mukai transform.

The structure sheaf of the diagonal ${\displaystyle {\mathcal {O}}_{\Delta }\in \mathrm {D} ^{b}(X\times X)}$, taken as a kernel, produces the identity functor on D^b(X). For a morphism f:X→Y, the structure sheaf of the graph Γ_f produces a pushforward when viewed as an object in D^b(X×Y), or a pullback when viewed as an object in D^b(Y×X).

Let ${\displaystyle X}$ be an abelian variety and ${\displaystyle {\hat {X}}}$ be its dual variety. The Poincaré bundle ${\displaystyle {\mathcal {P}}}$ on ${\displaystyle X\times {\hat {X}}}$, normalized to be trivial on the fiber at zero, can be used as a Fourier-Mukai kernel. Let ${\displaystyle p}$ and ${\displaystyle {\hat {p}}}$ be the canonical projections. The corresponding Fourier–Mukai functor with kernel ${\displaystyle {\mathcal {P}}}$ is then

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

There is a similar functor

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

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.

Let g denote the dimension of X. The Fourier–Mukai transformation is nearly involutive :

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

It interchanges Pontrjagin product and tensor product.

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

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

Deninger & Murre (1991) have used the Fourier-Mukai transform to prove the Künneth decomposition for the Chow motives of abelian varieties.

In string theory, T-duality (short for target space duality), which relates two quantum field theories or string theories with different spacetime geometries, is closely related with the Fourier-Mukai transformation.[2][3]

• Derived noncommutative algebraic geometry

• Deninger, Christopher; Murre, Jacob (1991), "Motivic decomposition of abelian schemes and the Fourier transform", J. Reine Angew. Math., 422: 201–219, MR 1133323

• Huybrechts, D. (2006), Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, vol. 1, The Clarendon Press Oxford University Press, doi:10.1093/acprof:oso/9780199296866.001.0001, ISBN 978-0-19-929686-6, MR 2244106
• Mukai, Shigeru (1981). "Duality between ${\displaystyle D(X)}$ and ${\displaystyle D({\hat {X}})}$ with its application to Picard sheaves". Nagoya Mathematical Journal. 81: 153–175. ISSN 0027-7630.