QUANTIZATION AND THE FOURIER-MUKAI TRANSFORM

DIMA ARINKIN

Here are the notes of my talk at the University of Chicago. I will post a better version when I have it.

1. Deformations

1.1. Deforming the Fourier-Mukai. The ground field is ℂ. Let A be a scheme and R an local Artinian ring with maximal ideal m_{R} such that R/m_{R} = ℂ. Recall that a deformation quantization of A is a sheaf of flat associative R-algebras Õ_{A} on A equipped with an identification Õ_{A}/m_{R}Õ_{A} = O_{A}. We consider Ã = (A, Õ_{A}) as a ringed space.

We will need a slightly more general kind of deformations (‘gerby deformations’); they are essentially stacks that are locally isomorphic to Ã/G_{m}, the precise definition will be discussed later. After we introduce the deformations we get the following (relatively simple) theorem (which is probably known to specialists.)

Theorem 1. Let A → B be a an abelian scheme and let Ã → B be its dual. To any deformation Ã of A, there corresponds a deformation Ã^{∨} of Ã^{∨} such that the Poincaré sheaf P on A×A^{∨} (it is supported by A×_{B} A^{∨}) extends to a flat R-module P̃ on Ã × (Ã^{∨})^{op} (that is, P̃/m_{R}P̃ = P).

Corollary 2. The Fourier-Mukai transform D(Ã) → D(Ã^{∨}) given by P̃ is an equivalence of categories.

Proof. (Skip this, probably.) Look at the adjoint functor. (Or just take square of Fourier — its kernel will be a deformation of the diagonal sheaf, so it is invertible.)

□

It is also easy to see that Ã^{∨} is determined by Ã up to a canonical isomorphism. Indeed, we claim that Ã^{∨} is the moduli space of line bundles with vanishing Chern class on fibers on Ã:

Definition 3. Let X̃ be a deformation quantization of a scheme X. By definition, a X̃-family of line bundles on fibers (vertical line bundles?) on Ã is a left quasicoherent sheaf l on X̃^{op} × Ã that is flat over OX̃ and such that l/lm_{R} is a line bundle on X ×_{B} A for some morphism X → B.

We say that l has Chern class 0 if restriction of l/lm_{R} to the fibers of X×_{B}A → X has Chern class 0.

Corollary 4. For a test scheme X and its deformation quantization X̃, the morphisms f : X̃ → Ã^{∨} are in one-to-one correspondence with X̃-families l of line bundles on fibers of Ã such that l has Chern class 0. The correspondence is f ↦ (f^{op} × id)^{∗}(P̃).

Date: 04/06/06.

Proof. The inverse correspondence goes like this: take adjoint (i.e., tensor rather than hom) Fourier transform of l. It is a sheaf l̂ on X̃^{op} × Ã^{∨} that is a flat over R, and l̂/m_{R}l̂ is the structure sheaf of the graph of a map X → A^{∨}. Therefore, l̂ is also the structure sheaf of the graph of a map, at least locally. This map is f. □

1.2. Proof of Theorem 1. We construct Ã^{∨} locally. Let U = Spec(S) ⊂ A^{∨} be an open set. Denote by M the Fourier transform of O_{U}; it is a quasi-coherent sheaf on A. We claim that there exists a deformation of M to a Ã-module M̃ that is Rflat, M̃ is unique up to an isomorphism, and S̃ := HomÃ(M̃, M̃) is a flat R-algebra such that S = S̃/m_{R}S̃. This is easy to see from deformation theory: obstructions to M̃ lie in Ext_{A}^{2}(M, M); different M̃’s are controlled by Ext_{A}^{1}(M, M), and lifts from S to S̃ are obstructed by Ext^{1}(M, M). However, all these vanish because they can be computed on the other side of Fourier transform.

Now we have constructed Ã^{∨} and P̃ locally. It remains to glue the pieces together using uniqueness of M̃ (that is, P̃). □

1.3. Gerby deformations. There’s a slight gap in the above proof (which explains where the gerbiness comes from). Namely, gluing is not completely trivial. We end up with the following object:

For a cover U_{i} = Spec(S_{i}) of A, we have a

(1) A non-commutative deformation of S_{i} to S̃_{i};

(2) On each intersection U_{ij} = Spec(S_{ij}) = U_{i}∩U_{j}, an isomorphism ϕ_{ij} between the nc-deformations of S_{ij} induced by S_{i} and S_{j}; (Here we assume for simplicity that A is separated.)

(3) On a triple intersection U_{ijk} = Spec(S_{ijk}), the automorphism of S̃_{ijk} induced by ϕ_{ij} ∘ϕ_{jk}∘ϕ_{ki} must be inner, moreover, there’s a chosen α_{ijk} ∈ S̃_{ijk} such that α_{ijk} = 1modm_{R} and the automorphism is conjugation by α_{ijk};

(4) α_{ijk}’s must satisfy a cocycle condition on intersections of four objects. This is exactly what we mean by a gerby deformation of A (of course, the inter-

esting part is writing what all these data can be changed by). A precise definition is written in Kontsevich.

The idea behind this definition is that if you have a gerby deformation, you have the notion of descent data for OÃ-modules. So even though A does not carry a sheaf of algebras, the category of modules over it still makes sense.

Example 5. Suppose all S̃_{i} are commutative. Then ϕ_{ij} satisfy the cocycle condition, so we get an honest (commutative) deformation Ã′ of A. The class α_{ijk} also satisfy the cocycle condition, so we get [α] ∈ H^{2}(Ã′, 1 + m_{R}OÃ′). In characteristic zero, can write [α] ∈ H^{2}(Ã′, m_{R}OÃ′).

So commutative gerby deformations can be thought of as a commutative deformation equipped with a gerbe cohomology class.

Remark 6. If the deformation is not commutative, there is no reason to expect Ã′ exists, and if it does, it need not be unique (it can be twisted by 1 + m_{R}OÃ′). So these gerbes are not gerbes over anything.

Example 7. A gerbe is trivial if α_{ijk} = 1. Then Ã is a usual deformation quantization. Another way to check for triviality is that in the category of sheaves on Ã (which makes sense even if it is gerby) the sheaf O_{A} admits a flat R-deformation E. We can then write OÃ = Hom(E, E). Notice that there might be more than one trivialization, and corresponding algebras need not be isomorphic.

2. First order deformations

Assume A is smooth. (And char is not 2?)

2.1. First order deformations splitting. Things simplify when R = ℂ[ℏ]/(ℏ^{2}). In this case, any gerby deformation is over something, and this something can be written as a ‘sum’ of something purely commutative and something purely quantum. More precisely, any deformation is a ‘sum’ of the following three kinds:

(1) Purely gerby: O_{A} does not change, but there is a non-trivial class [α] ∈ H^{2}(A, O_{A});

(2) Purely commutative: ‘usual’ deformations, classified by H^{1}(A, T_{A});

(3) Purely quantum: O_{A} does not change as a sheaf, but the product on it becomes a ⋆ b = ab + ℏτ(a, b). Classified by τ ∈ H^{0}(A, Λ^{2}T_{A}).

This splitting can be written as

(2.1)

HH^{2}(A) = H^{2}(A, O_{A}) ⊕ H^{1}(A, T_{A}) ⊕ H^{0}(A, Λ^{2}T_{A})
∥ ∥ ∥ ∥

First order deformations = Purely gerby

⊕

Purely commutative

⊕

Purely quantum

2.2. Duality on first order deformations. Recall now that π : A → B is an abelian scheme. π gives a filtration on T_{A}:

0 → T_{A/B} → T_{A} → π^{∗}T_{B} → 0;

this yields a filtration on exterior powers Λ^{k}T_{A}, and therefore on the terms of (2.1). For instance, H^{1}(A, T_{A}) fits into an exact sequence

H^{1}(A, T_{A/B}) → H^{1}(A, T_{A}) → H^{1}(A, π^{∗}T_{B}).

We can then write

H^{1}(A, π^{∗}T_{B}) = H^{1}(B, π_{∗}π^{∗}T_{B})⊕H^{0}(B, R^{1}π_{∗}π^{∗}T_{B}) = H^{1}(B, T_{B})⊕H^{0}(B, T_{B}⊗R^{1}π_{∗}T_{A/B}).

Applying this argument to all terms of (2.1), we get the following diagram:

[figure: H^{2}(O_{B}); ⊕; H^{1}(R^{1}π_{∗}O_{A}); ⊕; H^{0}(R^{2}π_{∗}O_{A}); H^{1}(π_{∗}T_{A/B}); ⊕; H^{0}(R^{1}π_{∗}T_{A/B})^{quot}; (H^{1}(T_{B}); ⊕; H^{0}(T_{B} ⊗ R^{1}π_{∗}O_{A}))^{sub}; H^{0}(π_{∗}Λ^{2}T_{A/B}); H^{0}(T_{B} ⊗ π_{∗}T_{A/B})^{sub}; H^{0}(Λ^{2}T_{B})^{sub}]

The rectangular blocks on the diagram correspond to terms of (2.1). Each term has a filtration; the corresponding subquotients are written vertically in the block. Finally, the superscripts (‘sub’ and ‘quot’) indicate that the subquotients is actually a subspace or a quotient space of the vector space in the diagram.

The spaces in the diagram admit the following interpretation:

gerbes on B

⊕

cocycles with values in P ic

⊕

[figure]

[figure]

[figure]

fiberwise gerbes

[figure]

[figure]

[figure]

deformations losing the zero section

⊕

deformation of the abelian scheme (B is fixed)

deformations

of B ⊕

deformations losing projection

π : A → B [figure]

[figure]

[figure]

fiberwise quantizations

quantizations which fix B

quantizations of B

As we pass from A to A^{∨}, we see non-trivial isomorphisms of the form R^{i}π_{∗}Λ^{j}T_{A/B} = R^{j}π_{∗}Λ^{i}T_{A^{∨}/B}. This is indicated by arrows in the above diagrams. They lead to the following interesting Fourier transforms:

(1) H^{0}(R^{1}π_{∗}T_{A/B}) = H^{0}(R^{1}π_{∗}T_{A^{∨}/B}): usual Fourier-Mukai; (if B is not fixed, it also includes the identity H^{1}(T_{B}) = H^{1}(T_{B}));

(2) H^{1}(R^{0}π_{∗}T_{A/B}) = H^{1}(R^{1}π_{∗}O_{A^{∨}}): Fourier-Mukai for torsors (follows from 1-motive picture, written in Donagi-Pantev);

(3) H^{0}(Λ^{2}T_{A/B}) = H^{0}(R^{2}π_{∗}O_{A^{∨}}): Ben-Bassat/Block/Pantev;

(4) H^{0}(R^{1}π_{∗}T_{A/B}) = H^{0}(T_{A/B} ⊗ T_{A}): The situation I was talking about last time (from my thesis).

3. Duality and properties

Now let us get back to deformations of order higher than one. What kinds of properties do they have?

Proposition 8. Ã^{∨} is not gerby if and only if Ã admits a ‘section’: that is, the structure sheaf of the zero section in A deforms.

Proof. Obvious: A^{∨} is not gerby if O_{A^{∨}} deforms. □

Proposition 9. Ã^{∨} is commutative if and only if locally on B, there exists a projection Ã → B̃ for a commutative B̃ (in particular, Ã is not gerby locally on B̃).

Proof. Suffices to show that Ã^{∨} is commutative if and only if locally on B there is a deformation of the zero section to a module Z̃ such that HomÃ^{∨}(Z̃, Z̃) is a deformation of B. The ‘only if’ direction is clear, the ‘if’ follows when one considers the Poisson bracket. □

Proposition 10. A projection Ã → B̃ means Ã^{∨} has a ‘two-sided section’ (if Ã or Ã^{∨} are gerby, need to be more careful).

In particular, suppose Ã^{∨} is commutative and not gerby. Then locally on B, we are exactly in the situation of the previous talk: Ã has a section and a projection. This is exactly the case when OÃ^{∨} has a tensor product, which translates into existence of convolution product (and so also a quantum groupoid structure) for Ã! To make things reasonable, assume Ã is not gerby, so Ã^{∨} admits a section. It is automatically a two-sided section, so we get a projection Ã → B̃ defined everywhere, not just locally. That’s what we did last time.