nLab
Fourier-Mukai transform

Context

Higher linear algebra

Motivic cohomology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Fourier-Mukai transform is a categorified integral transform roughly similar to the standard Fourier transform.

Generally, for X,YX,Y two suitably well-behaved schemes (e.g. affine, smooth, complex) and with D(X)D(X), D(Y)D(Y) their derived categories of quasicoherent sheaves, then a Fourier-Mukai transform with integral kernel ED(X×Y)E \in D(X\times Y) is a functor (of triangulated categories/stable (infinity,1)-categories)

Φ:D(X)D(Y) \Phi \colon D(X)\longrightarrow D(Y)

which is given as the composite of the (derived) operations of

  1. pull (inverse image) along the projection p X:X×YXp_X\colon X\times Y \to X

  2. tensor product with EE;

  3. push (direct image) along the other projection p Y:X×YYp_Y \colon X\times Y \to Y

i.e.

Φ(A)(p Y) *(Ep X *A) \Phi(A) \coloneqq (p_Y)_\ast (E\otimes p_X^\ast A)

(where here we implicitly understand all operations as derived functors). (e.g. Huybrechts 08, page 4)

Hence this is a pul-tensor-push integral transform through the product correspondence

X×Y X Y \array{ && X \times Y \\ & \swarrow && \searrow \\ X && && Y }

with twist EE on the correspondence space.

Such concept of integral transform is rather general and may be considered also in derived algebraic geometry (e.g. BenZvi-Nadler-Preygel 13) and lots of other contexts.

As discussed at integral transforms on sheaves this kind of integral transform is a categorification of an integral transform/matrix multiplication of functions induced by an integral kernel, the role of which here is played by ED(X×Y)E\in D(X \times Y).

Indeed, the central kind of result of the theory (theorem 1) says that every suitable linear functor D(X)D(Y)D(X)\to D(Y) arises as a Fourier-Mukai transform for some EE, a statement which is the categorification of the standard fact from linear algebra that every linear function between finite dimensional vector spaces is represented by a matrix.

The original Fourier-Mukai transform proper is the special case of the above where XX is an abelian variety, Y=A Y = A^\vee its dual abelian variety and EE is the corresponding Poincaré line bundle.

If XX is a moduli space of line bundles over a suitable algebraic curve, then a slight variant of the Fourier-Mukai transform is the geometric Langlands correspondence in the abelian case (Frenkel 05, section 4.4, 4.5).

Definition

Let XX and YY be schemes over a field KK. Let ED(QCoh(O X×Y))E \in D(QCoh(O_{X \times Y})) be an object in the derived category of quasi-coherent sheaves over their product. (This is a correspondence between XX and YY equipped with a chain complex EE of quasi-coherent sheaves).

The functor Φ(E):D(QCoh(O X))D(QCoh(O Y))\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y)) defined by

(1)FRq *(Lp *(F) LE), F \mapsto \mathbf{R}q_*(\mathbf{L}p^*(F) \otimes^{\mathbf{L}} E),

where pp and qq are the projections from X×YX \times Y onto XX and YY, respectively, is called the Fourier-Mukai transform of EE, or the Fourier-Mukai functor induced by EE.

When F:D(QCoh(O X))D(QCoh(O Y))F : D(QCoh(O_X)) \to D(QCoh(O_Y)) is isomorphic to Φ(E)\Phi(E) for some ED(QCoh(O X×Y))E \in D(QCoh(O_{X \times Y})), one also says that FF is represented by EE or simply that FF is of Fourier-Mukai type.

Properties

The key fact is as follows

Theorem (Orlov)

Let XX and YY be smooth projective varieties over a field KK. Let F:D(X)D(Y)F : D(X) \to D(Y) be a triangulated fully faithful functor. Then FF is represented by some object ED(X×Y)E \in D(X \times Y) which is unique up to isomorphism.

See Orlov 2003, 3.2.1 for a proof.

Remark

Though theorem 1 is stated there for FF admitting a right adjoint, it follows from Bondal-van den Bergh 2002 that every triangulated fully faithful functor admits a right adjoint automatically (see e.g. Huybrechts 08, p. 6).

Remark

It is generally believed that theorem 1 should be true for all triangulated functors (e.g. Huybrechts 08, p. 5).

Enhancements

On the level of the DG enhancements, it is true for all smooth proper KK-schemes that, in the homotopy category of DG categories, every functor corresponds bijectively to an isomorphism class of objects on D(X×Y)D(X \times Y). See (Toen 2006).

=–

References

Banerjee and Hudson have defined Fourier-Mukai functors analogously on algebraic cobordism.

  • Anandam Banerjee, Thomas Hudson, Fourier-Mukai transformation on algebraic cobordism, pdf.

Discussion of internal homs of dg-categories in terms of refined Fourier-Mukai transforms is in

  • Bertrand Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math. 167 (2007), 615–667

  • Alberto Canonaco, Paolo Stellari, Internal Homs via extensions of dg functors (arXiv:1312.5619)

Discussion in the context of geometric Langlands duality is in

For a discussion of Fourier-Mukai transforms in the setting of (,1)(\infty,1)-enhancements, see

Revised on January 22, 2015 10:54:35 by Urs Schreiber (88.100.66.95)