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 PD(X×Y)P \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)

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 September 8, 2014 05:35:26 by David Corfield (129.12.18.225)