Contents

cohomology

# Contents

## Idea

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

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

$\Phi \colon D(X)\longrightarrow D(Y)$

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

1. pull (inverse image) along the projection $p_X\colon X\times Y \to X$

2. tensor product with $E$;

3. push (direct image) along the other projection $p_Y \colon X\times Y \to Y$

i.e.

$\Phi(A) \coloneqq (p_Y)_\ast (E\otimes p_X^\ast A)$

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

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

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

with twist $E$ 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 $E\in D(X \times Y)$.

Indeed, the central kind of result of the theory (theorem ) says that every suitable linear functor $D(X)\to D(Y)$ arises as a Fourier-Mukai transform for some $E$, 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 $X$ is an abelian variety, $Y = X^\vee$ its dual abelian variety and $E$ is the corresponding Poincaré line bundle.

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

The functor $\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y))$ defined by

$F \mapsto \mathbf{R}q_*(\mathbf{L}p^*(F) \otimes^{\mathbf{L}} E),$

where $p$ and $q$ are the projections from $X \times Y$ onto $X$ and $Y$, respectively, is called the Fourier-Mukai transform of $E$, or the Fourier-Mukai functor induced by $E$.

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

## Properties

The key fact is as follows

###### Theorem (Orlov)

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

See Orlov 2003, 3.2.1 for a proof.

###### Remark

Though theorem is stated there for $F$ 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 was believed that theorem should be true for all triangulated functors (e.g. Huybrechts 08, p. 5). However according to (RVdB 2015) this is not true.

## Enhancements

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

• Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves. Nagoya Mathematical Journal 81: 153–175. (1981)

• Alexei Bondal, Michel van den Bergh. Generators and representability of functors in commutative and noncommutative geometry, 2002, arXiv

• Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172, translation.

• Lutz Hille, Michel van den Bergh, Fourier-Mukai transforms (arXiv:0402043)

• Daniel Huybrechts, Fourier-Mukai transforms, 2008 (pdf)

• C. Bartocci, Ugo Bruzzo, D. Hernandez Ruiperez, Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhauser 2009.

• Alice Rizzardo, Michel Van den Bergh, An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves (arXiv:1410.4039)

• Pieter Belmans, section 2.2 of Grothendieck duality: lecture 3, 2014 (pdf)

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 $(\infty,1)$-enhancements, see