On equivalences of derived categories of

coherent sheaves on abelian varieties

Dmitri Orlov

Introduction

Let A be an abelian variety and Â be the dual abelian variety. It was proved in [Mu1] that the derived categories of coherent sheaves D_{coh}^{b}(A) and D_{coh}^{b}(Â) are equivalent. This equivalence is called Fourier-Mukai transform. It is given with the help of the Poincare line bundle P_{A} on the product A × Â by the formula

F ↦ R^{·}p_{2∗}(P_{A} ⊗ p_{1}^{∗}(F)).

In the paper [Po1] this construction of Mukai is generalized in the following way. Consider two abelian varieties A and B . Let f be a homomorphism from A × Â to B × B̂ that is an isomorphism. It can be written as matrix

( \begin{matrix}α & β \\ γ & δ \end{matrix} )

where α is a homomorphism from A to B , β is from Â to B and etc.. We say that f is symplectic iff the inverse for f has the following matrix form

f^{−1} = ( \begin{matrix}δ̂ & −β̂ \\ −γ̂ & α̂ \end{matrix} )

It was proved in [Po1] that if the varieties A × Â and B × B̂ are symplectic isomorphic, where A and B are abelian varieties over algebraically closed field, then the derived categories of coherent sheaves D_{coh}^{b}(A) and D_{coh}^{b}(B) are equivalent.

The aim of this paper is to give the proof for the inverse statement (see Th.2.7 ), i.e. if the derived categories D_{coh}^{b}(A) and D_{coh}^{b}(B) are equivalent, then the varieties A × Â and B × B̂ are symplectic isomorphic. This theorem implies finiteness of the set of non-isomorphic abelian varieties, which have the derived categories of coherent sheaves equivalent to D_{coh}^{b}(A) (Cor.2.8 ). The proof is based on the theorem from [Or], which says that any exact equivalence between two derived categories of coherent sheaves on smooth projective varieties can be represented an object on the product.

Representing equivalences by objects on the product , we construct the map from the set of all exact equivalence between D_{coh}^{b}(A) and D_{coh}^{b}(B) to the set of symplectic isomorphism between A × Â and B × B̂ . It is proved that this map is functorial (Prop.2.4 ). In particular, we get the homomorphism from the group of exact autoequivalences of D_{coh}^{b}(A) to the group of the symplectic automorphisms of the variety A × Â . This homomorphism has the non-trivial kernel, which is isomorphic to the direct sum of ℤ and (A × Â)_{k} , where the latter is the group of k -points for the variety A × Â . If the field k is algebraically closed and chark = 0 , then we show that there is the short exact sequence

0 ⟶ ℤ ⊕ (A × Â)_{k} ⟶ Auteq(D_{coh}^{b}(A)) ⟶ Sp(A × Â) ⟶ 1

Further, we consider the central extension of Sp(A × Â) with ℤ and write the explicit formula for 2-cocycle that gives this extension. It can be done, because we know that any equivalence between derived categories of coherent sheaves on abelian varieties is represented by the object, which is isomorphic to a sheaf up to translation functor (Prop.2.9 ).

I am grateful to Max-Plank-Institut für Mathematik for hospitality and stimulating atmosphere.

1 Preliminaries.

Let X and Y be some smooth projective varieties over a field k . Denote by D_{coh}^{b}(X) and D_{coh}^{b}(Y) the bounded derived categories of coherent sheaves on X and Y respectively. Recall that a derived category has the structure of a triangulated category.

For every object E ∈ D_{coh}^{b}(Y × X) we can define an exact functor Φ_{E} from D_{coh}^{b}(X) to D_{coh}^{b}(Y) . Denote by p and π the projections of D_{coh}^{b}(X × Y) onto X and Y respectively:

X × Y ⟶^{π} Y
p↓
X

Then Φ_{E} is defined by the following formula:

Φ_{E}( ) := π_{∗}(E ⊗ p^{∗}( ))  (1)

(henceforth we abbreviate f_{∗}, f^{∗}, ⊗ and etc. instead of R f_{∗}, L f^{∗}, ⊗^{L} , because we consider only derived functors).

Moreover, the object E ∈ D_{coh}^{b}(X × Y) define another functor Ψ_{E} from D_{coh}^{b}(Y) to D_{coh}^{b}(X) . This functor is given by the formula:

Ψ_{E}( ) := p_{∗}(E ⊗ π^{∗}( ))  (2)

It is known that the functor Φ_{E} has left and right adjoint functors Φ_{E}^{∗} and Φ_{E}^{!} respectively, defined by the following formulas:

\begin{matrix}Φ_{E}^{∗} = Ψ_{E^{∨}⊗π^{∗}ω_{Y} [dimY]}, \\ Φ_{E}^{!} = Ψ_{E^{∨}⊗p^{∗}ω_{X}[dimX]}, \end{matrix}  (3)

where ω_{X} and ω_{Y} are the canonical sheaves on X and Y , and E^{∨} := R Hom(E, O_{X×Y}) .

Let X, Y, Z be smooth projective varieties and I, J, K objects of D_{coh}^{b}(X × Y) , D_{coh}^{b}(Y × Z) and D_{coh}^{b}(X × Z), respectively. Consider the following diagram of projections:

[figure: p_{12}; p_{13}; p_{23}; Y; XY Z; XZ; X; Z; XY; Y Z; π_{12}^{1}; π_{12}^{2}; π_{13}^{1}; π_{13}^{3}; π_{23}^{2}; π_{23}^{3}]

The objects I, J, K give the triple of functors

\begin{matrix}Φ_{I} : & D_{coh}^{b}(X) ⟶ D_{coh}^{b}(Y), \\ Φ_{J} : & D_{coh}^{b}(Y) ⟶ D_{coh}^{b}(Z), \\ Φ_{K} : & D_{coh}^{b}(X) ⟶ D_{coh}^{b}(Z), \end{matrix}

defined by the formula (1), i.e.

Φ_{I} := π_{12}^{2}_{∗}(I ⊗ π_{12}^{1}^{∗}(·)),
Φ_{J} := π_{23}^{3}_{∗}(J ⊗ π_{23}^{2}^{∗}(·)),
Φ_{K} := π_{13}^{3}_{∗}(K ⊗ π_{13}^{1}^{∗}(·)).

Let us consider the object I ⊠_{Y} J ∈ D_{coh}^{b}(X × Y × Z) . Henceforth, we will write I ⊠_{Y} J instead of p_{23}^{∗}J ⊗ p_{12}^{∗}I . The next proposition from [Mu1] gives us the composition law for exact fanctors between derived categories.

Proposition 1.1 The composition Φ_{J} ∘ Φ_{I} is isomorphic to Φ_{K} with

K = p_{13}_{∗}(I ⊠_{Y} J),  (4)

Thus, we associated to a smooth projective variety X the derived category of coherent sheaves D_{coh}^{b}(X) and we attached to each object E ∈ D_{coh}^{b}(X × Y) the functor Φ_{E} from D_{coh}^{b}(X) to D_{coh}^{b}(Y) .

There arise the following questions :

• When do two different smooth projective varieties have equivalent derived categories of coherent sheaves?

• What is the group of exact autoequivalences of the derived category of coherent sheaves for one X?

There are some previous results related to these questions. First, we can answer to these questions in the case of a variety with ample canonical or ample anticanonical sheaf.

Theorem 1.2 [BO2] Let X be a smooth projective variety with ample canonical or anticanonical sheaf. If D = D_{coh}^{b}(X) is equivalent as a triangulated category to D_{coh}^{b}(X′) for some other smooth algebraic variety X′ , then X is isomorphic to X′ .

Theorem 1.3 [BO2] Let X be a smooth projective variety with ample canonical or anticanonical sheaf. Then the group of isomorphism classes of exact autoequivalences D_{coh}^{b}(X) ⟶^{∼} D_{coh}^{b}(X) is generated by the automorphisms of the variety, the twists by invertible sheaves and the translations.

Moreover, we can add that in the hypothesis of Theorem 1.3 the group of autoequivalences AutD_{coh}^{b}(X) is the semi-direct product of the normal subgroup G_{1} = PicX ⊕ Z and the subgroup G_{2} = AutX naturally acting on G_{1} , i.e:

AuteqD_{coh}^{b}(X) ≅ AutX ⋉ (PicX ⊕ Z)

Secondly, It is known that any equivalence can be represented by an object on the product.

Theorem 1.4 [Or] Let X and Y be smooth projective varieties. Suppose F : D_{coh}^{b}(X) ⟶^{∼} D_{coh}^{b}(Y) is an exact equivalence. Then there exists a unique up to isomorphism object E ∈ D_{coh}^{b}(X × Y) such that the functors F and Φ_{E} are isomorphic.

Further, having an object E ∈ D_{coh}^{b}(X × Y) we need a good method to check fully faithfulness of the functor Φ_{E} : D_{coh}^{b}(X) ⟶ D_{coh}^{b}(Y) . It is not so simple to do. However, there is the useful criterion that will be needed for sequel.

Theorem 1.5 [BO1] Let M and X be smooth algebraic varieties over algebraically closed field of the characteristic 0 and let E ∈ D_{coh}^{b}(M × X) . Then the functor Φ_{E} is fully faithful, iff the following orthogonality conditions are verified:

i) Hom_{X}^{i}(Φ_{E}(O_{t_{1}}) , Φ_{E}(O_{t_{2}})) = 0  for every i and t_{1} ≠ t_{2}.
ii) Hom_{X}^{0}(Φ_{E}(O_{t}) , Φ_{E}(O_{t})) = k,
Hom_{X}^{i}(Φ_{E}(O_{t}) , Φ_{E}(O_{t})) = 0,  for i ∉ [0, dimM].

Here t , t_{1} , t_{2} are points of M and O_{t_{i}} are corresponding skyscraper sheaves.

Now, let X , Y , Z and W be four smooth projective varieties . Suppose we have two morphisms f and g :

f : Y × Z ⟶ Y and g : Y × Z ⟶ Z

such that the following condition holds:

(∗) The morphisms (f, pr_{2}) and (pr_{1}, g) from Y × Z to itself are isomorphisms.

The morphisms f and g give us the morphisms

(id_{X}, f) : X × Y × Z × W ⟶ X × Y and (g, id_{W}) : X × Y × Z × W ⟶ Z × W,

defined in the obvious way.

Let E and F be objects of the categories D_{coh}^{b}(X × Y) and D_{coh}^{b}(Z × W) respectively. Using the morphisms (f, pr_{2}) and (pr_{1}, g) we define the object H ∈ D_{coh}^{b}((X × Z) × (Y × W)) by the rule:

H := (id_{X}, f)^{∗}E ⊗ (g, id_{W})^{∗}F

Further, we can attach to the objects E , F , H the following functors:

Φ_{E} : D_{coh}^{b}(X) ⟶ D_{coh}^{b}(Y)
Φ_{F} : D_{coh}^{b}(Z) ⟶ D_{coh}^{b}(W)
Φ_{H} : D_{coh}^{b}(X × Z) ⟶ D_{coh}^{b}(Y × W)

defined by formula (1).

Proposition 1.6 In the above notation, if the functors Φ_{E} and Φ_{F} are full and faithful, then the functor Φ_{H} is full and faithful too. Proof. To prove the proposition we must show that the functor

Φ_{H}^{∗} ∘ Φ_{H} : D_{coh}^{b}(X × Z) ⟶ D_{coh}^{b}(X × Z),

which is the composition of the functor Φ_{H} with its left adjoint Φ_{H}^{∗} , is isomorphic to the identity functor id_{D_{coh}^{b}(X×Z)} . Let us consider the following diagram:

[figure: XZXZ; XZY W XZ; XZY XZ; XY XY; ZW Z; ZZ; XZY X; XY X; XX; XZX; (g, 1_{W} , g); (g, g); (1_{X}, f, 1_{X}, f); (1_{X}, f, 1_{X}); i_{Y}; i_{Z}; j_{Z}; q_{1}; q_{2}; q_{3}; q_{4}; q_{5}; q_{6}; 3; 1; 2; ⋆]

Here all q_{i} are projections, the morphisms i_{Y} , i_{Z}, j_{Z} are diagonal embeddings, i.e.

\begin{matrix}i_{Y} : & (x_{1}, y, x_{2}) & ↦ & (x_{1}, y, x_{2}, y) \\ i_{Z} : & (x_{1}, z, x_{2}) & ↦ & (x_{1}, z, x_{2}, z) \\ j_{Z} : & (x_{1}, z, y, x_{2}) & ↦ & (x_{1}, z, y, x_{2}, z) \end{matrix}

and the other morphisms are defined by the following rules:

\begin{matrix}(g, 1_{W} , g) : & (x_{1}, z_{1}, y, w, x_{2}, z_{2}) & ↦ & (g(y, z_{1}), w, g(y, z_{2})) \\ (g, g) : & (x_{1}, z_{1}, y, x_{2}, z_{2}) & ↦ & (g(y, z_{1}), g(y, z_{2})) \\ (1_{X}, f, 1_{X}, f) : & (x_{1}, z_{1}, y, x_{2}, z_{2}) & ↦ & (x_{1}, f(y, z_{1}), x_{2}, f(y, z_{2})) \\ (1_{X}, f, 1_{X}) : & (x_{1}, y, x_{2}, z) & ↦ & (x_{1}, f(y, z), x_{2}) \end{matrix}

In this diagram the square marked with ⋆ is commutative; all other squares are Cartesian. This is clear for squares 1 and 2, and this follows from condition (∗) for square 3.

Let us consider the following object

T := H ⊠_{Y ×W} H^{∨}⊗p_{Y ×W}^{∗} (ω_{Y ×W})[dim(Y ×W)] ∈ D_{coh}^{b}((X × Z) × (Y × W) × (X × Z))

To prove the lemma, it is sufficient to show that the object q_{2∗}q_{1∗}(T) is isomorphic to the structure sheaf of the diagonal Δ_{X×Z} ⊂ X × Z .

First, by the construction of H , the object T is isomorphic to

(g, 1_{W} , g)^{∗}(F ⊠_{W} F^{∨} ⊗ p_{W}^{∗} (ω_{W})[dimW]) ⊗ q_{1}^{∗}((1_{X}, f, 1_{X}, f)^{∗}(E ⊠ E^{∨}[dimY]) ⊗ p_{Y}^{∗} (ω_{Y}))

Hence, by the projection formula, the object q_{1∗}(T) is isomorphic to

q_{1∗}(g, 1_{W} , g)^{∗}(F ⊠_{W} F^{∨} ⊗ p_{W}^{∗} (ω_{W})[dimW]) ⊗ (1_{X}, f, 1_{X}, f)^{∗}(E ⊠ E^{∨}[dimY]) ⊗ p_{Y}^{∗} (ω_{Y})

Further, by the theorem of flat base change [H], we get the following isomorphism:

q_{1∗}(g, 1_{W} , g)^{∗}(F ⊠_{W} F^{∨} ⊗ p_{W}^{∗} (ω_{W})[dimW]) ≅ (g, g)^{∗}q_{3∗}(F ⊠_{W} F^{∨} ⊗ p_{W}^{∗} (ω_{W})[dimW]).

By assumption, the functor Φ_{F} is full and faithful. Consequently, the object q_{3∗}(F ⊠_{W} F^{∨}⊗p_{W}^{∗} (ω_{W})[dimW]) is isomorphic to O_{Δ_{Z}} . Besides, it immediately follows from condition (∗) that

(g, g)^{∗}O_{Δ_{Z}} ≅ j_{Z∗}O_{XZY X}

Therefore, we get the following isomorphism:

q_{1∗}(T) ≅ j_{Z∗}O_{XZY X} ⊗ (1_{X}, f, 1_{X}, f)^{∗}(E ⊠ E^{∨}[dimY]) ⊗ p_{Y}^{∗} (ω_{Y})

Further, using the projection formula again, we obtain

q_{1∗}(T) ≅ j_{Z∗}(j_{Z}^{∗}(1_{X}, f, 1_{X}, f)^{∗}(E ⊠ E^{∨}[dimY]) ⊗ p_{Y}^{∗} (ω_{Y}))

Since the square marked with ⋆ is commutative, we may write:

q_{1∗}(T) ≅ j_{Z∗}((1_{X}, f, 1_{X})^{∗}i_{Y}^{∗} (E ⊠ E^{∨}[dimY]) ⊗ p_{Y}^{∗} (ω_{Y})) ≅

j_{Z∗}((1_{X}, f, 1_{X})^{∗}(E ⊠_{Y} E^{∨}[dimY]) ⊗ p_{Y}^{∗} (ω_{Y}))

Since the morphism (f, pr_{2}) : Y × Z ⟶ Y × Z is an isomorphism, we have (f, pr_{2})^{∗}ω_{Y ×Z} ≅ ω_{Y ×Z} . This implies that f^{∗}(ω_{Y}) ⊗ pr_{2}^{∗}(ω_{Z}) ≅ pr_{1}^{∗}(ω_{Y}) ⊗ pr_{2}^{∗}(ω_{Z}) . Hence we get

f^{∗}(ω_{Y}) ≅ pr_{1}^{∗}(ω_{Y})  (5)

Therefore, we can write

q_{1∗}(T) ≅ j_{Z∗}((1_{X}, f, 1_{X})^{∗}(E ⊠_{Y} E^{∨} ⊗ p_{Y}^{∗} (ω_{Y})[dimY]))

Applying the functor q_{2∗} to the object q_{1∗}(T) and using commutativity of diagram 2, we get

q_{2∗}q_{1∗}(T) ≅ i_{Z∗}q_{5∗}(1_{X}, f, 1_{X})^{∗}(E ⊠_{Y} E^{∨} ⊗ p_{Y}^{∗} (ω_{Y})[dimY])

Further, by the theorem of flat base change, we have

q_{2∗}q_{1∗}(T) ≅ i_{Z∗}q_{6}^{∗}q_{4∗}(E ⊠_{Y} E^{∨} ⊗ p_{Y}^{∗} (ω_{Y})[dimY])

By assumption, the functor Φ_{E} is full and faithful. This yields that

q_{4∗}(E ⊠_{Y} E^{∨} ⊗ p_{Y}^{∗} (ω_{Y})[dimY]) ≅ O_{Δ_{X}}

Finally, we get

q_{2∗}q_{1∗}(T) ≅ i_{Z∗}q_{6}^{∗}(O_{Δ_{X}}) ≅ O_{Δ_{X×Z}}

This completes the proof of this proposition. □

Corollary 1.7 In the above notation, if the functors Φ_{E} and Φ_{F} are equivalences, then the functor Φ_{H} is an equivalence too. Proof. By the proposition, the functor Φ_{H} is full and faithful. Moreover, the left adjoint functor Φ_{H}^{∗} is coincides with

Ψ_{H^{∨}⊗p_{Y ×W}^{∗} (ω_{Y ×W})}[dimY × W]

Using the equality (5) we obtain

H^{∨} ⊗ p_{Y ×W}^{∗} (ω_{Y ×W}) ≅ (id_{X}, f)^{∗}(E^{∨} ⊗ p_{Y}^{∗} (ω_{Y})) ⊗ (g, id_{W})^{∗}(F^{∨} ⊗ p_{W}^{∗} (ω_{W}))

Therefore, by the proposition, the functor Φ_{H}^{∗} is full an faithful too. This yields that Φ_{H} is an equivalence. □

Notice that f = pr_{1} and g = pr_{2} satisfy the condition (*). Consequently, we have the following corollary.

Corollary 1.8 Suppose E and F are objects of D_{coh}^{b}(X × Y) and D_{coh}^{b}(Z × W) respectively such that the functors Φ_{E} and Φ_{F} are full and faithful. Then the functor

Φ_{H} : D_{coh}^{b}(X × Z) ⟶ D_{coh}^{b}(Y × W),

where H := E ⊠ F , is full and faithful too.

2 Derived categories of abelian varieties.

Let A be an abelian variety. Denote by m_{A} the multiplication morphism m_{A} : A × A ⟶ A . From now on, we write the group low in A additively. Moreover we will use the following notations:

– we denote by d_{A} : A × A ⟶ A the composition m_{A} ∘ (1_{A}, (−1)_{A}) , i.e. d_{A} : (a_{1}, a_{2}) ↦ a_{1} − a_{2} ;

– for a closed point a ∈ A , by T_{a} : A⊗_{k} k(a) ⟶ A⊗_{k} k(a) we denote the translation morphism T_{a}(b) = a + b ;

– and the dual abelian variety of A will be denoted by Â := Pic^{0}A . Any closed k -point a′ ∈ Â defines the line bundle P_{a′} ∈ Pic^{0}A on A . Moreover, if we have some abelian varieties A_{1}, ..., A_{m} and a k -point (a′_{1}, ..., a′_{m}) ∈ Â_{1} × ⋯ × Â_{m} , then we will use the notation P_{(a′_{1},...,a′_{k})} for the line bundle P_{a′_{1}} ⊠⋯⊠P_{a′_{k}} on the product A_{1} × ⋯ × A_{k} .

It was proved by Mukai in [Mu1] that the derived categories of coherent sheaves D_{coh}^{b}(A) and D_{coh}^{b}(Â) are equivalent. He considered the functor Φ_{P_{A}} , where P_{A} is a Poincare line bundle on A×Â , and showed that this functor gives an equivalence between D_{coh}^{b}(A) and D_{coh}^{b}(Â) .

Let us consider two abelian varieties A and B . By P_{A} and P_{B} denote Poincare line bundles on A × Â and B × B̂ respectively. Suppose the categories D_{coh}^{b}(A) and D_{coh}^{b}(B) are equivalent as triangulated categories. By F denote a functor from D_{coh}^{b}(A) to D_{coh}^{b}(B) that gives some equivalence. By Theorem 1.4 , we can find an object E ∈ D_{coh}^{b}(A × B) such that it represents this equivalence. In other words, there is some object E such that the functor F is isomorphic to the functor Φ_{E} defined by formula (1).

Let us consider an object E ∈ D_{coh}^{b}(A × B) and construct the objects H ∈ D_{coh}^{b}(A × Â × A × B) and G ∈ D_{coh}^{b}(A × B × B × B̂) by the following rule:

\begin{matrix}H := & p_{23}^{∗}P_{A} ⊗ (m_{A}, p_{4})^{∗}E \\ G := & p_{24}^{∗}P_{B}^{∨} ⊗ (p_{1}, d_{B})^{∗}E^{∨} \end{matrix}  (6)

Note that, by Proposition 1.6 , if Φ_{E} is an equivalence, then the functors

\begin{matrix}Φ_{H} : D_{coh}^{b}(A × Â) ⟶ D_{coh}^{b}(A × B) \\ Φ_{G} : D_{coh}^{b}(A × B) ⟶ D_{coh}^{b}(B × B̂) \end{matrix}

are equivalences too.

The composition of these functors Φ_{G} ∘ Φ_{H}[n] is a functor from D_{coh}^{b}(A × Â) to D_{coh}^{b}(B × B̂) . Denote by J ∈ D_{coh}^{b}(A × Â × B × B̂) the object that represents this composition, i.e. Φ_{J} ≅ Φ_{G} ∘ Φ_{H}[n] . The object J can be constructed by the rule (4), i.e.

J ≅ p_{1256∗}(H ⊠_{A×B} G)[n]  (7)

In this way, we attached to each object E ∈ D_{coh}^{b}(A × B) the object J ∈ D_{coh}^{b}(A × Â × B × B̂) so that if the functor Φ_{E} is an equivalence, then the functor Φ_{J} is an equivalence too. Now we would like to show that if the functor Φ_{E} is an equivalence then the object J is an invertible sheaf on some subvariety that is a graf of some isomorphism between A × Â and B × B̂ .

But, at first, we prove the following lemma that is needed for sequel. Let X̄ := X ×_{k} k̄ and denote by F̄ the inverse image of F under the morphism X̄ ⟶ X .

Lemma 2.1 Let F be a coherent sheaf on a smooth variety X . Suppose there exist a closed subvariety j : Z ↪ X̄ and an invertible sheaf L on Z such that F̄ ≅ j_{∗}L . Then there exist a closed subvariety i : Y ↪ X and an invertible sheaf M on Y such that F ≅ i_{∗}M and j = ī . Proof. The subvariety Y is the image of the composition Z ⟶^{j} X̄ ⟶ X . The question is local on X . Let us consider a closed point y ∈ Y ⊂ X . Denote by (A, m) the local ring of X at the point y . Then, F_{y} is the module over A . We must prove that F_{y} ≅ A/J , i.e there exists a surjective homomorphism A ⟶ F_{y} → 0 . By Nakayama’s lemma, it is sufficient to show that F_{y}/mF_{y} ≅ A/mA . Since A/mA is a field, we have F_{y}/mF_{y} ≅ (A/mA)^{s} .

Now let us take a closed point z ∈ Z ⊂ X̄ over the point y . And let (B, n) be the local ring of X̄ at z . Further , we have F̄_{z}/mF̄_{z} ≅ (B/mB)^{s} . Hence, we get F̄_{z}/nF̄_{z} ≅ (B/nB)^{s} . This implies that s = 1 . Thus, F_{y} ≅ A/J for some ideal J and F̄_{z} ≅ B/JB = A/J ⊗_{A} B . Lemma is proved. □

Remark It follows from the proof that the analogous statement is true not only for field extensions but for any surjective morphism.

The following lemma says us that the property to be full and faithful is stable under field extensions.

Lemma 2.2 Let X and Y be smooth projective variety over field k and let E be an object of D_{coh}^{b}(X × Y) . Then the functor Φ_{E} : D_{coh}^{b}(X) ⟶ D_{coh}^{b}(Y) is full and faithful iff the functor ΦĒ : D_{coh}^{b}(X̄) ⟶ D_{coh}^{b}(Ȳ) is full and faithful. Proof. ⇒ Denote by Φ_{E}^{∗} the left adjoint functor for Φ_{E} . If the functor Φ_{E} is full and faithful, then the composition Φ_{E} ∘ Φ_{E}^{∗} is the identity functor id_{D_{coh}^{b}(X)} , which is represented by the structure sheaf of the diagonal O_{Δ} on the product X × X . Hence, by the theorem of flat base change, the composition ΦĒ ∘ Φ∗ ; Ē is represented the structure sheaf OΔ̄ , where Δ̄ is the diagonal in X̄ × X̄ . Consequently, the functor ΦĒ is full and faithful too.

⇐ Consider the composition Φ_{E} ∘ Φ_{E}^{∗} . It is represented by an object J on X × X . There exists the canonical morphism ϕ : J ⟶ O_{Δ} . Since the functor ΦĒ is full and faithful, the morphism ϕ̄: J̄ ⟶ OΔ̄ is an isomorphism. This implies that ϕ is an isomorphism too. Thus, the functor Φ_{E} is full and faithful. □

Proposition 2.3 In the above notation, suppose that the functor Φ_{E} is an equivalence. Then there exists an abelian subvariety j : X ↪ A × Â × B × B̂ such that the canonical homomorphisms X ⟶ A × Â and X ⟶ B × B̂ are isomorphisms, and J ≅ j_{∗}L , where L is an invertible sheaf on X . Proof. By Lemmas 2.1 and 2.2 , we can assume that k is algebraically closed.

Denote by e and e′ the identity points of A × Â and B × B̂ respectively. We know that Φ_{H}(O_{e}) ≅ E . Moreover, the quasi-inverse for the functor Φ_{G}[n] is the functor Ψ_{G^{∨}} (see (3)) and Ψ_{G^{∨}}(O_{e′}) ≅ E . Therefore, since Φ_{J} ≅ Φ_{G} ∘ Φ_{H}[n] we get Φ_{J} (O_{e}) ≅ O_{e′} . Hence we have

J ⊗ O_{{e}×(B×B̂)} ≅ O_{{e}×{e′}}

This implies that there exists a Zariski neighborhood U of e such that the restriction J |_{U×(B×B̂)} is a sheaf with the support which meets {e} × (B × B̂) only at the point {e} × {e′} .

Denote by (R, m) and (R′, m′) the local rings of A × Â and B × B̂ at the points e and e′ respectively. Then (R̃, m̃) := R ⊗_{k} R′ is the local ring of the product at the point {e} × {e′} . By J denote the stalk of the sheaf J |_{U×(B×B̂)} at the point {e} × {e′} . It is a module over ring R̃ .

Further, we know that

(R/m ⊗_{k} R′) ⊗R̃ J = k and TorR̃ ; i(R/m ⊗_{k} R′, J) = 0 for all i > 0

Consequently, we obtain J/m̃J ≅ R̃/m̃R̃ . By Nakayma’s lemma , we have a surjective homomorphism R̃ ⟶ J → 0 . Since R̃ is a flat algebra over R , we get

R/m ⊗_{R} J = k and Tor_{i}^{R}(R/m, J) = 0 for all i > 0

This immediately yields that J is a free R -module of rang 1 and the composition R ⟶ R̃ ⟶ J is isomorphism. Consequently, there exist Zariski neighborhood V of e and subscheme X(V) ⊂ V × (B × B̂) such that the projection X(V) ⟶ V is an isomorphism and J |_{V ×(B×B̂)}≅ O_{X(V)} .

Now, consider a closed point (a, a′) ∈ V . We have

Φ_{J} (O_{(a,a′)}) ≅ O_{(b,b′)}

for some closed point (b, b′) ∈ B × B̂ . Further, it follows from the construction of the objects H and G that

Φ_{H}(O_{(a,a′)}) ≅ T_{(a,0)}^{∗}E ⊗ P_{(a′,0)} and Ψ_{G^{∨}}(O_{(b,b′)}) ≅ T_{(0,−b)}^{∗}E ⊗ P_{(0,b′)}

Hence, we obtain

T_{(a,0)}^{∗}E ⊗ P_{(a′,0)} ≅ T_{(0,−b)}^{∗}E ⊗ P_{(0,b′)}

Now take an arbitrary closed point (a, a′) ∈ A × Â . We can represent it as a sum (a, a′) = (a_{1}, a′_{1}) + (a_{2}, a′_{2}) with (a_{1}, a′_{1}), (a_{2}, a′_{2}) ∈ V . There exists the closed points (b_{1}, b′_{1}) and (b_{2}, b′_{2}) such that

T_{(a_{i},0)}^{∗}E ⊗ P_{(a′_{i},0)} ≅ T_{(0,−b_{i})}^{∗}E ⊗ P_{(0,b′_{i})}

for i = 1, 2 . Hence, we have

T_{(a,0)}^{∗}E ⊗ P_{(a′,0)} ≅ T_{(0,−b)}^{∗}E ⊗ P_{(0,b′)}

for the point (b, b′) that is the sum of (b_{1}, b′_{1}) and (b_{2}, b′_{2}) . Consequently, we get

Φ_{J} (O_{(a,a′)}) ≅ O_{(b,b′)}

for an arbitrary closed point (a, a′) . Thus, repeating the procedure stated above, we can find for every closed point (a, a′) a neighborhood W and a subscheme X(W) ⊂ W ×(B ×B̂) such that the projection X(W) ⟶ W is an isomorphism and J |_{W ×(B×B̂)}≅ O_{X(W)} . Finally, there exist an abelian variety j : X ↪ A × Â× B × B̂ such that the canonical morphism X ⟶ A × Â is an isomorphism and J ≅ j_{∗}L , where L is an invertible sheaf on X . The subvariety X gives the homomorphism from A×Â to B ×B̂ , which induces an equivalence of derived categories of coherent sheaves. Hence, this homomorphism is an isomorphism. The proposition is proved. □

Remark. It follows from the proposition that if Φ_{E} is an equivalence, then E gives the isomorphism f_{E} : A × Â ⟶ B × B̂ with the graf X . We see that f_{E}(a, a′) = (b, b′) iff T_{(a,0)}^{∗}E ⊗ P_{(a′,0)} ≅ T_{(0,−b)}^{∗}E ⊗ P_{(0,b′)} .

For two abelian varieties A and B denote by Eq(A, B) the set of all exact equivalences, up to isomorphism, from D_{coh}^{b}(A) to D_{coh}^{b}(B) .

Let us introduce two categories 𝔄 and 𝔇 , which are groupoids (i.e. all morphisms are isomorphisms). Objects of both categories are abelian varieties. Morphisms in 𝔄 are homomorphisms between abelian varieties, which are isomorphisms; morphisms in 𝔅 are exact equivalences between derived categories of coherent sheaves on abelian varieties, i.e.

\begin{matrix}mor_{𝔄}(A, B) := Iso(A, B) \\ mor_{𝔇}(A, B) := Eq(A, B) \end{matrix}

By Proposition 2.3 , we have a map from the set Eq(A, B) to the set Iso(A × Â, B × B̂) , which attaches to each equivalence Φ_{E} the isomorphism f_{E} . Consider the map F : 𝔇 ⟶ 𝔄 that takes an abelian variety A to A × Â and acts on morphisms as above, i.e

F (Φ_{E}) = f_{E}

Proposition 2.4 In the above notation, the map F : 𝔇 ⟶ 𝔄 is a functor. Proof. We can assume that field k is algebraically closed. Consider three abelian variety A, B, C . Let E and F be objects from D_{coh}^{b}(A × B) and D_{coh}^{b}(B × C) respectively such that the functors

\begin{matrix}Φ_{E} : D_{coh}^{b}(A) ⟶ D_{coh}^{b}(B) \\ Φ_{F} : D_{coh}^{b}(B) ⟶ D_{coh}^{b}(C) \end{matrix}

are equivalence. By I denote the object p_{13∗}(E ⊠_{B} F) in D_{coh}^{b}(A × C) .

We must show that

f_{I} = f_{F} ∘ f_{E}

It is sufficient to check for closed point. We know that f_{E}(a, a′) = (b, b′) for (a, a′) ∈ A × Â and (b, b′) ∈ B × B̂ iff T_{(a,0))}^{∗}E ⊗ P_{(a′,0)} ≅ T_{(0,−b)}^{∗}E ⊗ P_{(0,b′)} .

Now fix point (a, a′) ∈ A × Â . Let f_{E}(a, a′) = (b, b′) and f_{F}(b, b′) = (c, c′) . We must check that f_{I}(a, a′) = (c, c′) .

First, we have the following sequence of isomorphisms:

T_{(a,0)}^{∗}I ⊗ P_{(a′,0)} ≅ p_{13∗}((T_{(a,0)}^{∗}E ⊗ P_{(a′,0)}) ⊠_{B} F) ≅ p_{13∗}((T_{(0,−b)}^{∗}E ⊗ P_{(0,b′)}) ⊠_{B} F)

Further, it is clear that for any object S ∈ D_{coh}^{b}(A × B × C) we have

p_{13∗}(T_{b}^{∗}S) ≅ p_{13∗}(S)

Consequently, we obtain

T_{(a,0)}^{∗}I ⊗ P_{(a′,0)} ≅ p_{13∗}(E ⊠_{B}(T_{(b,0)}^{∗}F ⊗ P_{(b′,0)})) ≅

p_{13∗}(E ⊠_{B}(T_{(0,−c)}^{∗}F ⊗ P_{(0,c′)})) ≅ T_{(0,−c)}^{∗}I ⊗ P_{(0,c′)}

This yields f_{I}(a, a′) = (c, c′) and the proposition is proved. □

Thus, we have got the functor F : 𝔇 ⟶ 𝔄 . Now we would like to describe it. First, not every element from Iso(A × Â, B × B̂) can be realized as f_{E} for some E .

Let us consider a morphism f : A × Â ⟶ B × B̂ . It can be written as matrix

( \begin{matrix}α & β \\ γ & δ \end{matrix} )

where α is the morphism from A to B , β is from Â to B and etc. Any morphism f determines two morphisms f̂ and f̃ from B × B̂ to A × Â . They have the following matrix forms:

f̂ = ( \begin{matrix}δ̂ & β̂ \\ γ̂ & α̂ \end{matrix} );  f̃ = ( \begin{matrix}δ̂ & −β̂ \\ −γ̂ & α̂ \end{matrix} )

Let us introduce the set Sp(A × Â, B × B̂) . It is a subset of Iso(A × Â, B × B̂) consisting of all f such that f̃ coincides with inverse for f , i.e.

Sp(A × Â, B × B̂) := {f ∈ Iso(A × Â, B × B̂) | f̃ = f^{−1}}

(Note that Sp(A × Â) is a subgroup of Aut(A × Â) ).

Definition 2.5 We say that a morphism f : A × Â ⟶ B × B̂ is symplectic if it belongs to Sp(A × Â, B × B̂) .

Proposition 2.6 For any equivalence Φ_{E} : D_{coh}^{b}(A) ⟶ D_{coh}^{b}(B) the morphism f_{E} is symplectic. Proof. First, going over to the algebraic closure, we can assume that field k is algebraically closed. To check the equality f̃ = f^{−1} it is sufficient to consider the closed points. Let f takes (a, a′) ∈ A × Â to (b, b′) ∈ B × B̂ . We must show that f̂(b, −b′) = (a, −a′) .

The isomorphism f_{E} is given by the abelian subvariety X ↪ A × Â × B × B̂ . Consequently, we have to check that P_{(−a′,a,b′,−b)} ⊗ O_{X} ≅ O_{X} . It is equivalent to show that P_{(−a′,a,b′,−b)} ⊗ J ≅ J .

From the beginning let us consider the object H = p_{23}^{∗}P_{A} ⊗ (m_{A}, p_{4})^{∗}E from D_{coh}^{b}(A × Â × A × B) (see (6)). We have the following sequence of isomorphisms:

P_{(−a′,a,0,0)} ⊗ H ≅ p_{23}^{∗}(P_{A} ⊗ P_{(a,0)}) ⊗ (m_{A}, p_{4})^{∗}E ⊗ P_{(−a′,0,0,0)} ≅
p_{23}^{∗}(T_{(0,a)}^{∗}P_{A}) ⊗ (m_{A}, p_{4})^{∗}E ⊗ P_{(−a′,0,0,0)} ≅
T_{(0,0,a,0)}^{∗}(p_{23}^{∗}P_{A} ⊗ T_{(0,0,−a,0)}^{∗}((m_{A}, p_{4})^{∗}E) ⊗ P_{(−a′,0,0,0)}) ≅
T_{(0,0,a,0)}^{∗}(p_{23}^{∗}P_{A} ⊗ (m_{A}, p_{4})^{∗}(T_{(−a,0)}^{∗}E ⊗ P_{(−a′,0)}) ⊗ P_{(0,0,a′,0)}) ≅
T_{(0,0,a,0)}^{∗}(p_{23}^{∗}P_{A} ⊗ (m_{A}, p_{4})^{∗}(T_{(0,b)}^{∗}E ⊗ P_{(0,−b′)}) ⊗ P_{(0,0,a′,0)}) ≅
T_{(0,0,a,b)}^{∗}(p_{23}^{∗}P_{A} ⊗ (m_{A}, p_{4})^{∗}(E)) ⊗ P_{(0,0,a′,−b′)} ≅ T_{(0,0,a,b)}^{∗}H ⊗ P_{(0,0,a′,−b′)}

Further, we have the similar sequence of isomorphisms for the object G := p_{24}^{∗}P_{B}^{−1} ⊗ (p_{1}, d_{B})^{∗}E^{∨} from D_{coh}^{b}(A × B × B × B̂) :

G ⊗ P_{(0,0,b′,−b)} ≅ p_{24}^{∗}(P_{B}^{−1} ⊗ P_{(0,−b)}) ⊗ (p_{1}, d_{B})^{∗}E^{∨} ⊗ P_{(0,0,b′,0)} ≅

p_{24}^{∗}(T_{(b,0)}^{∗}P_{B}^{−1}) ⊗ (p_{1}, d_{B})^{∗}E^{∨} ⊗ P_{(0,0,b′,0)} ≅

T_{(0,b,0,0)}^{∗}(p_{24}^{∗}P_{B}^{−1} ⊗ T_{(0,−b,0,0)}^{∗}((p_{1}, d_{B})^{∗}E^{∨}) ⊗ P_{(0,0,b′,0)}) ≅

T_{(0,b,0,0)}^{∗}(p_{24}^{∗}P_{B}^{−1} ⊗ (p_{1}, d_{B})^{∗}(T_{(0,−b)}^{∗}E^{∨} ⊗ P_{(0,−b′)}) ⊗ P_{(0,b′,0,0)}) ≅ T_{(0,b,0,0)}^{∗}(p_{24}^{∗}P_{B}^{−1} ⊗ (p_{1}, d_{B})^{∗}(T_{(a,0)}^{∗}E^{∨} ⊗ P_{(−a′,0)}) ⊗ P_{(0,b′,0,0)}) ≅ T_{(a,b,0,0)}^{∗}(p_{24}^{∗}P_{B}^{−1} ⊗ (p_{1}, d_{B})^{∗}(E^{∨})) ⊗ P_{(−a′,b′,0,0)} ≅ T_{(a,b,0,0)}^{∗}G ⊗ P_{(−a′,b′,0,0)}

Finally, we know that the object J is isomorphic to p_{1256∗}(H ⊠_{A×B} G) . Hence, combining the previous calculations, we get

P_{(−a′,a,b′,−b)} ⊗ J ≅ p_{1256}_{∗}(T_{(0,0,a,b,0,0)}^{∗}(H ⊠_{A×B} G)) ≅ J .

This proves the proposition. □

Propositions 2.3 and 2.6 implies the following theorem Theorem 2.7 Let A and B be two abelian varieties. If the derived categories of coherent sheaves D_{coh}^{b}(A) and D_{coh}^{b}(B) are equivalent as triangulated categories, then A × Â and B × B̂ are symplectic isomorphic.

The inverse statement was prove in [Po1] for algebraically closed field. In the next chapter we will give a little different proof of it.

Corollary 2.8 For each abelian variety A there are only finite number of nonisomorphic abelian varieties , which have the derived categories of coherent sheaves equivalent to D_{coh}^{b}(A) (as triangulated categories). Proof. It was proved in [LOZ] that for each abelian variety Z there are, up to isomorphism, only finitely many abelian varieties that admit an embedding in X as an abelian subvarieties. This statement for Z = A × Â and Theorem 2.7 imply our statement. □

Further, it follows from Propositions 2.4 and 2.6 that we have homomorphism from the group of exact equivalence Auteq(D_{coh}^{b}(A)) to the group of symplectic automorphisms Sp(A×Â) . Before to describe the kernel and the image of this homomorphism we show that if the functor Φ_{E} is an equivalence then E have to be a sheaf, up to translation. Note that this fact is not valid in general, i.e. there are examples of equivalences, which are represented by complexes, having more than one cohomology.

Proposition 2.9 Let E be an object in D_{coh}^{b}(A × B) such that the functor Φ_{E} : D_{coh}^{b}(A) ⟶^{∼} D_{coh}^{b}(B) is an exact equivalence. Then E has only one cohomology, i.e. it is isomorphic to a sheaf up to translation. Proof. Let us consider the following commutative diagram:

[figure: (A × Â) × (A × B) × (B × B̂); A × B × A × B; (A × Â) × (B × B̂); A × B; p; g; f; q]

Here p and q are projections and the morphisms f and g are defined by the following rules:

\begin{matrix}f : & (a_{1}, a′, a_{2}, b_{1}, b_{2}, b′) & ↦ & (a_{1} + a_{2}, b_{1}, a_{2}, b_{1} − b_{2}) \\ g : & (a_{1}, b_{1}, a_{2}, b_{2}) & ↦ & (a_{1} − a_{2}, b_{1} − b_{2}). \end{matrix}

Now take the object

K := (H ⊠_{A×B} G) ⊗ p_{56}^{∗}P_{B}[2n],

which belongs to the derived category of coherent sheaves on (A × Â) × (A × B) × (B × B̂) . It follows from definition of H and G (see (6)) that

f_{∗}K ≅ f_{∗}(p_{23}^{∗}P_{A} ⊗ p_{46}^{∗}P_{B}^{−1} ⊗ p_{56}^{∗}P_{B}) ⊗ (E ⊠ E^{∨})[2n] ≅ O_{M} ⊗ (E ⊠ E^{∨}).

It is easily to see that M is A×B×{0}×{0} as abelian subvariety of A×B×A×B . Therefore, we get

g_{∗}f_{∗}K ≅ E ⊗ (E^{∨} |_{(0,0)}),

where E^{∨} |_{(0,0)} is a complex of vector spaces that is the inverse image of E^{∨} under the closed embedding of the point (0, 0) into A × B .

On the other side, it follows from Proposition 2.3 that

p_{∗}K ≅ j_{∗}L

where j is an embedding of the abelian variety X into A × Â × B × B̂ that gives the isomorphism f_{E} between A × Â and B × B̂ , and L is an invertible sheaf on X . Denoting by q′ the restriction q on X , we obtain the isomorphism

g_{∗}f_{∗}K ≅ q′_{∗}L

We know that q′ is a morphism of abelian varieties. Let us denote by d the dimension of Ker(q′) . Then dim Im(q′) = 2n − d and the cohomology sheaves H^{j}(q′_{∗}L) are trivial for j ∉ [0, d] .

Now we assume that the last right non-trivial cohomology of E is H^{0}(E) . Let H^{−i}(E) with i ≥ 0 be the last left non-trivial cohomology.

Supports of all cohomology sheaves of the object E are subsets of Im(q′) . This implies that codim Supp H^{−i}(E) ≥ d . Consequently, all cohomologies of the object (H^{−i}(E))^{∨}[−i] , which have index less than i + d , are trivial.

Further, we have the canonical homomorphism H^{−i}(E)[i] ⟶ E . Hence, there exists a non-trivial homomorphism

E^{∨} ⟶ (H^{−i}(E))^{∨}[−i]

We know that all non-trivial cohomology of the second object lie in [i+d, ∞] . Hence, there exists j ≥ i + d such that H^{j}(E^{∨}) ≠ 0 . Replacing E with T_{(a,b)}^{∗}E , if it is necessary, we can assume that the point (0, 0) is contained in Supp H^{j}(E^{∨}) . Consequently, the object

q′_{∗}L ≅ g_{∗}f_{∗}K ≅ E^{∨} |_{(0,0)} ⊗E  (8)

has non-trivial cohomology for the same j ≥ i + d . Hence , we get i = 0 and the object E has only one non-trivial cohomology H^{0}(E) . Thus, it is isomorphic to a sheaf. □

Let us put B = A . Suppose E is a sheaf on A × A such that Φ_{E} is an equivalence and f_{E} is the identity, i.e. the graf X is the diagonal of (A × Â) × (A × Â) . By Δ : A ⟶ A × A denote the diagonal embedding. It follows from (8) that there exists an object W on A such that Δ_{∗}(W) ≅ E^{∨} |_{(0,0)} ⊗E .

First, assume that the point (0, 0) belongs to Supp E . This implies that E^{∨} |_{(0,0)} is non-trivial complex of vector spaces. Hence, there exists some sheaf E on A such that Δ_{∗}(E) ≅ E . Therefore, we have Φ_{E}(·) ≅ E ⊗ (·) . Since Φ_{E} is an equivalence, E is an invertible sheaf. It is easily to check that E ∈ Pic^{0}(A) , because f_{E} is the identity.

If now the point (0, 0) does not belong to Supp E , we take a sheaf E′ := T_{(a_{1},a_{2})}^{∗}E such that its support contains (0, 0) . It is clear that f_{E′} = f_{E} . As above, we get E′ ≅ Δ_{∗}(E′) with E′ ∈ Pic^{0}(A) . Hence, E ≅ T_{(a_{1}−a_{2},0)}^{∗}Δ_{∗}(E′) . Thus, we have proved the following corollary.

Corollary 2.10 Let A be an abelian variety. Then the kernel of the homomorphism Auteq(D_{coh}^{b}(A)) ⟶ Sp(A × Â) is isomorphic to the group ℤ ⊕ (A × Â)_{k} , where (A × Â)_{k} denotes the group of k -points of the abelian variety A × Â .

3 Semi-homogeneous vector bundles.

In the previous chapter we showed that the morphism f_{E} is symplectic. Now, using the technique of [Mu2] and results from [BO1], we are going to prove that every symplectic morphism f : A × Â ⟶ B × B̂ can be realized the same way. This fact can be deduced from [Po1] and [Po2]. It was proved in the paper [Po1] that if the variety A × Â and B × B̂ are symplectically isomorphic, then the derived categories D_{coh}^{b}(A) and D_{coh}^{b}(B) are equivalent. We will give a little different proof for this statement.

Henceforth, we assume that k is an algebraically closed field and char(k) = 0 .

Definition 3.1 A vector bundle on an abelian variety Z is semi-homogeneous if for every z ∈ Z , there exists a line bundle L on Z such that T_{z}^{∗}(E) ≅ E ⊗ L . (Note that L has to belong to Pic^{0}(Z) ).

The following theorem was proved in [Mu2].

Proposition 3.2 ( [Mu2], Th.5.8.) Let E be a simple vector bundle on Z . Then the following conditions are equivalent:

(1) dim H^{j}(Z, End(E)) = \binom{n}{j} for all j = 0, ..., n,
(2) E is semi-homogeneous,
(3) End(E) is a homogeneous vector bundle,
(4) there exist an isogeny π : Y ⟶ Z and a line bundle L on Y such that
E ≅ π_{∗}(L)

Let E be a vector bundle on Z . We denote by μ(E) the equivalence class of \frac{det(E)}{r(E)} in NS(Z) ⊗_{ℤ} ℚ .

It is well-known that any line bundle L on Z gives the homomorphism ϕ_{L} : Z ⟶ Ẑ and this mapping induces the canonical injection of NS(Z) into Hom(Z, Ẑ) ( see, for example, [Mum]). Further, every element μ = \frac{[L]}{l} ∈ NS(Z) ⊗_{ℤ} ℚ defines the correspondence Φ_{μ} ⊂ Z × Ẑ . It is clear that Φ_{μ} = Im[Z ⟶^{(l,ϕ_{L})} Z × Ẑ] . Denote by q_{1} and q_{2} the projections of Φ_{μ} onto Z and Ẑ respectively.

The following proposition was proved in [Mu2].

Proposition 3.3 ([Mu2], Th.7.10.) Let μ = \frac{[L]}{l} , where [L] is the equivalence class in NS(Z) of a line bundle L and l is a positive integer.

(1) There exists a simple semi-homogeneous vector bundle E with μ(E) = μ
(2) Every simple semi-homogeneous vector bundle E′ with μ(E′) = μ

is isomorphic to E ⊗ M for some line bundle M in Pic^{0}(Z)

(3) r(E)^{2} = deg(q_{1}), and χ(E)^{2} = deg(q_{2})

It is easy to show that for any point (z, z′) ∈ Φ_{μ} we have isomorphism T_{z}^{∗}(E) ≅ E ⊗ P_{z′} . Actually, let l = r(E) and L = det(E) . We know that (z, z′) = (lx, ϕ_{L}(x)) for some point x ∈ Z . Moreover, since E is semi-homogeneous, we have T_{x}^{∗}(E) ≅ E ⊗ M for some line bundle M in Pic^{0}(Z) . Taking determinants, we obtain T_{x}^{∗}(L) ≅ L ⊗ M^{⊗l} . It means that P_{ϕ_{L}(x)} = M^{⊗l} . On the other side, we have T_{lx}^{∗}(E) ≅ E ⊗ M^{⊗l} . Since (z, z′) = (lx, ϕ_{L}(x)) , we get T_{z}^{∗}(E) ≅ E ⊗ P_{z′} .

Now let us introduce a closed subgroup Σ^{0}(E) ⊂ Ẑ defined by the following rule

Σ^{0}(E) := {z′ ∈ Ẑ | E ⊗ P_{z′} ≅ E}

If a bundle E is simple semi-homogeneous, then by Proposition 3.2 , there exist an isogeny π : Y ⟶ Z and a line bundle M on Y such that E ≅ π_{∗}(M) . It is clear that π^{∗}(E) ≅ M_{1} ⊕ M_{2} ⊕ ⋯ ⊕ M_{r} for some line bundles M_{i} on Y with M_{1} = M . If z′ ∈ Σ^{0}(E) then π^{∗}(P_{z′}) ⊗ M ≅ M_{i} for i = 1, .., r . This yields that ordΣ^{0}(E) ≤ r^{2} . On the other side, we have q_{2∗}(Ker(q_{1})) ⊂ Σ^{0}(E) . Hence, we obtain ordΣ^{0}(E) = r^{2} and q_{2∗}(Ker(q_{1})) = Σ^{0}(E) .

Further, let us suppose that T_{z}^{∗}(E) ≅ E ⊗ P_{z′} for some point (z, z′) ∈ Z × Ẑ . Since the homomorphism Φ_{μ} ⟶ Z is an isogeny, there exists a point z′′ ∈ Ẑ such that (z, z′′) ∈ Φ_{μ} . Hence, we obtain E ⊗ P_{(z′−z′′)} ≅ E . Consequently, (z′ − z′′) ∈ Σ^{0}(E) . But we have proved that q_{2∗}(Ker(q_{1})) = Σ^{0}(E) . This immediately implies that (0, z′ − z′′) ∈ Φ_{μ} . Therefore, we get (z, z′) ∈ Φ_{μ} . Thus, we have proved

Lemma 3.4 In the above notation, if E is a simple semi-homogeneous vector bundle on Z . Then T_{z}^{∗}(E) ≅ E ⊗ P_{z′} iff (z, z′) ∈ Φ_{μ} .

Now we are ready to prove that if f ∈ Sp(A×Â, B ×B̂) then there exists a sheaf E on A × B such that f_{E} = f .

Let us fix an symplectic isomorphism f : A × Â ⟶ B × B̂ . Let X be the graf of f . Isomorphism f can be written as matrix

f = ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

We can assume that β : Â ⟶ B is an isogeny, because every f can be represented as a product of f_{1} ∈ Sp(A × Â, B × B̂) and f_{2} ∈ Sp(A × Â) such that β_{1} and β_{2} are isogenies.

Since β is an isogeny, we can attach to f the element g ∈ Hom(A × Â, B × B̂) ⊗_{ℤ} ℚ that has the following matrix form:

g = ( \begin{matrix}β^{−1}α & −β \\ −β̂^{−1} & δβ^{−1} \end{matrix} )

As f is symplectic, it can easily be checked that ĝ = g . There exists the canonical injection of NS(A × B) ⊗_{ℤ}ℚ into Hom(A × Â, B × B̂) ⊗_{ℤ}ℚ and it is known that an element g belongs to the image of this injection iff ĝ = g (see [Mum]). This yields that there exists μ = \frac{[L]}{l} ∈ NS(A × B) such that Φ_{μ} coincides with the graf of the correspondence g . By Proposition 3.3 , we can find a simple semi-homogeneous bundle E on A × B with μ(E) = μ .

We affirm that Φ_{E} is an equivalence and f_{E} = f . Before to prove it let us compare X and Φ_{μ} , which are the grafs of f and g . If a point (a, a′, b, b′) is contained in X , then

\begin{aligned}b & = & α(a) + β(a′) \\ b′ & = & γ(a) + δ(a′) \end{aligned} and, consequently \begin{matrix}a′ = & −β^{−1}α(a) & + & β^{−1}(b) \\ b′ = & (γ − δβ^{−1}α)(a) & + & δβ^{−1}(b) \end{matrix}

Since f is symplectic, we have the equality (γ − δβ^{−1}α) = β̂^{−1} . This follows that (a, a′, b, b′) belongs to X iff (a, −a′, b, b′) belongs to Φ_{μ} . Consequently, Φ_{μ} = (1_{A}, −1Â, 1_{B}, 1B̂)X .

Proposition 3.5 In the above notation, the functor Φ_{E} : D_{coh}^{b}(A) ⟶ D_{coh}^{b}(B) is an equivalence. Proof. Let us denote by E_{a} the restriction of the bundle E on {a} × B . By Theorem 1.5 , to prove that Φ_{E} is full and faithful we must check that E_{a} are simple and they are orthogonal to each other for different points.

First, notice that the rank of the bundle E is equal to \sqrt{deg(β}) . This follows from Proposition 3.3 and from the fact that the degree of the homomorphism X ⟶ A × B is equal to deg(β) .

It is clear that E_{a} is semi-homogeneous (see (4) of Prop.3.2 ) and μ(E_{a}) = δβ^{−1} ∈ NS(B) ⊗ ℚ ⊂ Hom(B, B̂) ⊗ ℚ . By ν denote δβ^{−1} as the element of NS(B) ⊗ ℚ . By Proposition 3.3 , there exists a simple semi-homogeneous vector bundle F on B with μ(F) = ν . It is easily to see that Φ_{ν} = Im[Â ⟶^{(β,δ)} B × B̂] . Since f is an isomorphism, the homomorphism Â ⟶^{(β,δ)} B × B̂ is an embedding. Hence, we obtain r(F) = \sqrt{deg(β)} = r(E_{a}) .( Moreover, we can see that the canonical projection Σ^{0}(E) ⟶ Σ^{0}(E_{a}) is an isomorphism.) The bundles F and E_{a} are semihomogeneous and they have the same slope μ ∈ NS(B) and the same rank. Moreover, F is simple. Hence, E_{a} is simple too (see [Mu2] 6.18).

Further, since E is semi-homogeneous, we have E_{a} ≅ E_{0} ⊗ M for some M ∈ Pic^{0}(B) . It follows from condition (4) of Proposition 3.2 and the projection formula that either there exists non-trivial morphism E_{0} ⟶ E_{0} ⊗ M or they are orthogonal, i.e.

Hom^{i}(E_{0}, E_{0} ⊗ M) = 0, and Hom^{i}(E_{0} ⊗ M, E_{0}) = 0 for all i .

But, it is known that a simple semi-homogeneous vector bundle is stable (see [Mu2] 6.15.). Hence, we have either E_{0} and E_{0} ⊗ M are isomorphic or they are orthogonal.

Now let us assume that E_{0} is isomorphic to E_{a} for some point a ∈ A . Since E is semi-homogeneous, we have T_{(a,0)}^{∗}E ≅ E ⊗ P_{(a′,b′)} for some point (a′, b′) ∈ Â × B̂ . This implies

E_{a} ⊗ P_{b′} ≅ E_{0} ≅ E_{a}

Hence, P_{b′} ∈ Σ^{0}(E_{a}) . As the morphism Σ^{0}(E) ⟶ Σ^{0}(E_{a}) is an isomorphism, there exists a′′ ∈ Â such that E ⊗ P_{(a′′,b′)} ≅ E . Therefore, we obtain T_{(a,0)}^{∗}E ≅ E ⊗P_{(a′−a′′,0)} . Hence, by Lemma 3.4 , we obtain (a, a′ −a′′, 0, 0) ∈ Φ_{μ} . On the other hand, as f is an isomorphism, the projection Φ_{μ} ⟶ A × Â is an isomorphism too. This immediately implies a = 0 . Thus, E_{a_{1}} and E_{a_{2}} are orthogonal for any two different points a_{1} and a_{2} . By Theorem 1.5 , the functor Φ_{E} : D_{coh}^{b}(A) ⟶ D_{coh}^{b}(B) is full and faithful. By the same reason the adjoint functor Ψ_{E^{∨}} is full and faithful too. This yields that Φ_{E} is an equivalence. The proposition is proved. □

By Proposition 2.3 , the equivalence Φ_{E} gives the isomorphism f_{E} : A × Â ⟶ B × B̂ and f_{E}(a, a′) = (b, b′) iff T_{(a,b)}^{∗}E ≅ E ⊗ P_{(−a′,b′)} . Hence, if Y is the graf of f_{E} , then Y = (1_{A}, −1Â, 1_{B}, 1B̂)Φ_{μ} for μ = μ(E) . Therefore, the graf f_{E} coincides with the graf of f , i.e f_{E} = f .

Thus, we have proved the following theorems

Theorem 3.6 Let A and B be abelian varieties over algebraically closed field k with char(k) = 0 . Then the derived categories of coherent sheaves D_{coh}^{b}(A) and D_{coh}^{b}(B) are equivalent as triangulated categories iff there exists a symplectic isomor-

phism f : A × Â ⟶^{∼} B × B̂

Theorem 3.7 Let A be an abelian variety over algebraically closed field k with chark = 0 . Then the group of exact autoequivalence, up to isomorphism, Auteq(D_{coh}^{b}(A)) can be included into the short exact sequence

0 ⟶ ℤ ⊕ (A × Â)_{k} ⟶ Auteq(D_{coh}^{b}(A)) ⟶ Sp(A × Â) ⟶ 1

Let us describe the group Auteq(D_{coh}^{b}(A)) more precisely. It has the normal subgroup (A×Â)_{k} , which consists of the functors T_{a}^{∗}(·)⊗P_{a′} with (a, a′) ∈ A×Â . The quotient group is a central extension of Sp(A × Â) with ℤ . We denote this central extension by S̃p(A × Â) . We have the short exact sequences:

0 ⟶ (A × Â)_{k} ⟶ Auteq(D_{coh}^{b}(A)) ⟶ S̃p(A × Â) ⟶ 1  (9)
0 ⟶ ℤ ⟶ S̃p(A × Â) ⟶ Sp(A × Â) ⟶ 1  (10)

To describe the central extension (10) it is sufficient to give a 2-cocycle λ(g_{1}, g_{2}) ∈ ℤ , where g_{1} and g_{2} are elements of Sp(A × Â) .

By Proposition 2.9 , if Φ_{E} is an equivalence, then E ∈ D_{coh}^{b}(A × A) is isomorphic to E[k] for some sheaf E on A × A . Let E and F be some sheaf on A × A such that Φ_{E} and Φ_{F} are equivalence. The composition Φ_{F} ∘ Φ_{E} is represented by the object G[k] , where G is a sheaf. Let us put λ(f_{F} , f_{E}) = k . It is obvious that λ is 2-cocycle on the group Sp(A × Â) .

We can calculate λ explicitly. Suppose k = ℂ . Then A ≅ ℂ^{n}/ℤ^{2n} and we can associated with any line bundle L on A the Hermitian form H(L) on ℂ^{n} (see [Mum]). Denote by p(H) the number of positive eigenvalues of H . Thus, we get the function p : NS(A) → ℤ . We can extend it to NS(A) ⊗ ℝ by the rule: p(r[L]) = p(H(L)) for r ∈ ℝ_{+} . Let us take two elements of Sp(A × Â) :

g_{1} = ( \begin{matrix}α_{1} & β_{1} \\ γ_{1} & δ_{1} \end{matrix} ) and g_{2} = ( \begin{matrix}α_{2} & β_{2} \\ γ_{2} & δ_{2} \end{matrix} )  (11)

such that β_{1} and β_{2} are isogenies. This means that there exist the inverses β_{1}^{−1}, β_{2}^{−1} ∈ Hom(A, Â) ⊗ ℚ . Further, let us write:

g_{1}g_{2} = ( \begin{matrix}α_{3} & β_{3} \\ γ_{3} & δ_{3} \end{matrix} )

Now consider the element β_{1}^{−1}β_{3}β_{2}^{−1} = β_{1}^{−1}α_{1}+δ_{2}β_{2}^{−1} of Hom(A, Â)⊗ℚ . It belongs to NS(A) ⊗ ℚ , hence it can be written as \frac{[L]}{l} for some line bundle L on A and natural integer l . It is not hard to prove that

λ(g_{1}, g_{2}) = p(β_{1}^{−1}β_{3}β_{2}^{−1}) − n.

Thus, we have given the formula for λ(g_{1}, g_{2}) , when β_{1} and β_{2} are invertible. Since λ is a cocycle, it is uniquely determined by this formula. Moreover, we see that our cocycle can be extend onto the group Sp(A × Â, ℝ) ⊂ End(A × Â) ⊗ ℝ .

Now suppose A = E^{n} , where E is an elliptic curve without complex multiplication, then the group Sp(A × Â) is isomorphic to Sp_{2n}(ℤ) . Consider the real symplectic group G = Sp_{2n}(ℝ) . It is known that the Maslov index defines the ℤ - valued 2-cocycle on the symplectic group by the rule:

μ_{l}(g_{1}, g_{2}) = τ(l, g_{1}l, g_{1}g_{2}l),

where l is some Lagrangian plane, and τ is the Maslov index (see [LV]). There is the formula for the cocycle μ . If g_{1} and g_{2} are written as matrices like in (11), then

μ_{l}(g_{1}, g_{2}) = sign(β_{1}^{−1}β_{3}β_{2}^{−1}),

where the right side denotes the signature of symmetric matrix β_{1}^{−1}β_{3}β_{2}^{−1} . We can compare the cocycles λ and μ . It is easy to check that cocycle 2λ − μ is trivial as element of the second cohomology , i.e it gives the splitting extension. The symplectic group G = Sp_{2n}(ℝ) has the universal covering group G̃ . It is known that the central extension G̃_{μ} , which is given by the cocycle μ , is included in the exact sequence 1 → G̃ → G̃_{μ} → ℤ/4ℤ ⟶ 0 . Hence, we have the exact sequence 1 → G̃ → G̃_{λ} → ℤ/2ℤ → 0 for G̃_{λ} . It is easily to show that this sequence splits. Therefore, G̃_{λ} is isomorphic to G̃ × ℤ/2ℤ .

Example. Let A be an abelian variety such that End(A) = ℤ . Then the Neron-Severy group NS(A) is isomorphic to ℤ . Let L and M be a generators of NS(A) and NS(Â) respectively such that ϕ_{M} ∘ ϕ_{L} = N · id_{A} with N > 0 . In this case the group Sp(A × Â) is isomorphic to congruenz-subgroup Γ_{0}(N) ⊂ SL(2, ℤ) . Further, if B is an abelian variety such that B ×B̂ is isomorphic to A×Â , then B can be represented as the image of some morphism A ⟶^{k·id,mϕ_{L}} A × Â . We can assume that g.c.d.(k, m) = 1 . Denote by ψ this morphism from A to B . The kernel of ψ is equal to Ker(mϕ_{L}) ∩ A_{k} . Since g.c.d.(k, m) = 1 , we get Ker(ψ) = Ker(ϕ_{L}) ∩ A_{k} . Besides, we know that Ker(ϕ) ⊂ A_{N} . Hence, we can assume that k is a divisor of N . Thus, any k , which is a divisor of N , gives us the abelian variety B , where B := A/(Ker(ϕ_{L}) ∩ A_{k}) . It is clear that two different divisors determine nonisomorphic abelian varieties. Moreover, it is easily to prove that the embedding B into A × Â is split iff g.c.d.(k, \frac{N}{k}) = 1 . Therefore, the number of abelian varieties B such that D_{coh}^{b}(B) ≃ D_{coh}^{b}(A) is equal to 2^{s} , where s is the number of the prime divisors of N .

Let us assume, in addition, that A is principal polarized. So we have N = 1 and if D_{coh}^{b}(A) ≃ D_{coh}^{b}(B) for an abelian variety B , then B ≅ A . Moreover, the group Sp(A × Â) is isomorphic to SL(2, ℤ) . Further, it was proved in [Po2] that the sequence (9) splits for principal polarized abelian variety. Finally, we get the description of Auteq(D_{coh}^{b}(A)) as semi-direct product of normal subgroup (A × Â)_{k} and the group S̃p(A × Â) , i.e.

Auteq(D_{coh}^{b}(A)) ≅ S̃p(A × Â) ⋉ (A × Â)_{k}

and S̃p(A × Â) is the central extension of SL(2, ℤ) , which is determined by cocycle λ . We can include it in the following exact sequence:

1 ⟶ B_{3} ⟶ S̃p(A × Â) ⟶ ℤ/2nℤ,

where B_{3} is the braid group of 3 threads. ( B_{3} is the central extension of SL(2, ℤ) , which is induced by the universal covering of SL(2, ℝ)) .

References

[BO1] A. I. Bondal, D. O. Orlov, Semiorthogonal decomposition for algebraic varieties, Preprint MPI/95-15 (see also alg-geom9506012).

[BO2] A. I. Bondal, D. O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Preprint MPI/97-36.

[H] R. Hartshorne, Residues and duality, Lecture Notes in Math., vol.20, Springer-Verlag, 1966.

[LV] G. Lion, M. Vergne, The Weil representation, Maslov index and theta series, Birkhäuser, 1980.

[Mu1] S. Mukai, Duality between D(X) and D(X̂) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), pp.153–175.

[Mu2] S. Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ., v.18, N2 (1978), pp.239-272.

[Mum] D. Mumford, Abelian varieties, Oxford Univ. Press, Second Edition, 1974.

[LOZ] H. W. Lenstra, Jr. F. Oort, Yu. G. Zarhin, Abelian subvarieties, J. Algebra, v.180,(1996), pp.513-516.

[Or] D. Orlov, Equivalences of derived categories and K3 surfaces, Preprint MPI/97-45. ( to be published in Journal of Math. Sciences, Plenum Publ. Corp.,Vol.34, Alg. geom.-5, vol.85, No.6, (1997)).

[Po1] A. Polishchuk, Symplectic biextensions and generalization of the Fourier-Mukai transforms Math. Research Letters , v.3 (1996), pp.813-828.

[Po2] A. Polishchuk, Biextensions, Weil representations on derived categories, and theta-functions Ph.D. Thesis, Harvard University, 1996.

Algebra section, Steklov Math. Institute, Vavilova 42, 117333 Moscow, RUSSIA orlov@genesis.mi.ras.ru