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| In algebraic geometry, the '''Fourier–Mukai transform''' or '''Mukai–Fourier transform''', introduced by {{harvs|txt|last=Mukai|year=1981|authorlink=Shigeru Mukai}}, is an isomorphism between the [[derived categories]] of [[coherent sheaves]] on an [[abelian variety]] and its dual. It is analogous to the classical [[Fourier transform]] that gives an isomorphism between [[Distribution (mathematics)|tempered distribution]]s on a finite-dimensional real vector space and its dual.
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| If the [[Canonical bundle|canonical class]] of a variety is positive or negative, then the derived category of coherent sheaves determines the variety. The Fourier–Mukai transform gives examples of different varieties (with trivial canonical bundle) that have isomorphic derived categories, as in general an abelian variety of dimension greater than 1 is not isomorphic to its dual.
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| ==Definition==
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| Let <math>X</math> be an [[abelian variety]] and <math>\hat X</math> be its [[Dual abelian variety|dual variety]]. We denote by <math>\mathcal P</math> the [[Poincaré bundle]] on
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| :<math>X \times \hat X,</math>
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| normalized to be trivial on the fibers at zero. Let <math>p</math> and <math>\hat p</math> be the canonical projections.
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| The Fourier–Mukai functor is then
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| :<math>R\mathcal S: \mathcal F \in D(X) \mapsto R\hat p_\ast (p^\ast \mathcal F \otimes \mathcal P) \in D(\hat X)</math>
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| The notation here: ''D'' means [[derived category]] of [[coherent sheaves]], and ''R'' is the [[higher direct image functor]], at the derived category level.
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| There is a similar functor
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| :<math>R\widehat{\mathcal S} : D(\hat X) \to D(X). \, </math>
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| ==Properties==
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| Let ''g'' denote the dimension of ''X''.
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| The Fourier–Mukai transformation is nearly involutive :
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| :<math>R\mathcal S \circ R\widehat{\mathcal S} = (-1)^\ast [-g]</math>
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| It transforms [[Pontrjagin product]] in [[tensor product]] and conversely.
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| :<math>R\mathcal S(\mathcal F \ast \mathcal G) = R\mathcal S(\mathcal F) \otimes R\mathcal S(\mathcal G)</math>
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| :<math>R\mathcal S(\mathcal F \otimes \mathcal G) = R\mathcal S(\mathcal F) \ast R\mathcal S(\mathcal G)[g]</math>
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| ==References==
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| *{{Citation | last1=Huybrechts | first1=D. | title=Fourier–Mukai transforms in algebraic geometry | url=http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001 | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-929686-6; 978-0-19-929686-6 | doi=10.1093/acprof:oso/9780199296866.001.0001 | id={{MR|2244106}} | year=2006}}
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| *{{cite journal
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| | last=Mukai
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| | first=Shigeru
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| | authorlink=Shigeru Mukai
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| | title=Duality between <math>D(X)</math> and <math>D(\hat X)</math> with its application to Picard sheaves
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| | journal=Nagoya Mathematical Journal
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| | volume=81
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| | year=1981
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| | pages=153–175
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| | id=ISSN 0027-7630
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| | url=http://projecteuclid.org/euclid.nmj/1118786312
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| }}
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| {{DEFAULTSORT:Fourier-Mukai transform}}
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| [[Category:Abelian varieties]]
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