Reconstruction of a variety from the derived

category and groups of autoequivalences.

A. Bondal  D. Orlov

Introduction.

There exist examples of different varieties X having equivalent the derived categories D_{coh}^{b}(X) of coherent sheaves. For abelian varieties and K3 surfaces this kind of equivalences were constructed by Mukai, Polishchuk and the second author in [Mu1], [Mu2], [P], [Or]. In [BO] we prove equivalence of the derived categories for varieties connected by some kind of flops.

Does it mean that D_{coh}^{b}(X) is a weak invariant of a variety X ? In this paper we shall show that this is not the case at least for some types of algebraic varieties.

We prove that a variety X is uniquely determined by its category D_{coh}^{b}(X) , if its anticanonical (Fano case) or canonical (general type case) sheaf is ample.

To reconstruct a variety from the category we, in fact, use nothing but a graded structure of the category, i.e. we only need to fix the translation functor.

The idea is that for good, in the above sense, varieties we can recognize the skyscraper sheaves of closed points in D_{coh}^{b}(X) . The main tool for this is the Serre functor [BK] (see also ch.1), which for D_{coh}^{b}(X) can be regarded as a categorical incarnation of the canonical sheaf ω_{X} .

With respect to the above problem it is natural to introduce the groupoid with object being the categories D_{coh}^{b}(X) and with morphisms being autoequivalences.

There are two natural questions related to a groupoid: which objects are isomorphic and what is the group of automorphisms of an individual object. The first problem was considered above in the framework of graded categories. To tackle the second one we need triangulated structure of the category. In chapter 3 we prove that for a smooth algebraic variety with ample either canonical or anticanonical sheaf the group of exact autoequivalences is the semidirect product of the group of automorphisms of the variety and the Picard group plus translations.

The answers to the above questions for the case of varieties with non-ample and non-antiample canonical sheaf seem to be of considerable interest.

We are thankful to H.Meltzer for a stimulating discussion and to Max-Plank-Institut für Mathematik for hospitality.

1 Preliminaries.

We collect here some facts relating to functors in graded and triangulated categories, especially those about Serre functor.

In this paper for simplicity reasons we consider only k -linear additive categories, where k is an arbitrary field.

By definition a graded category is a couple (D, T_{D}) consisting of a category D and a fixed equivalence functor T_{D} : D ⟶ D , called translation functor.

Recall that a triangulated category is a graded category with an additional structure: a distinguished class of exact triangles satisfying certain axioms (see [V]).

A functor F : D ⟶ D′ between two graded categories D and D′ is called graded if it commutes with the translation functor. More precisely, there is fixed a natural isomorphism of functors:

t_{F} : F ∘ T_{D} →^{∼} T_{D′} ∘ F.

In the sequel we omit subscripts in the notation of translation functors because from their position in formulas it is always clear to which category they belong.

While considering graded functors, we use graded natural transformations. A natural transformation μ between graded functors F and G is called graded if the following diagram is commutative:

F ∘ T →^{t_{F}} T ∘ F
↓μT  ↓T μ
G ∘ T →^{t_{G}} T ∘ G.

A graded functor F : D ⟶ D′ between triangulated categories is called exact if it transforms all exact triangles into exact triangles in the following sense. If X → Y → Z → T X is an exact triangle in D , then one takes F X → F Y → F Z → F T X and substitutes in this sequence F T (X) by T F (X) using the natural isomorphism of F T with T F . The result

F X → F Y → F Z → T F X

should be an exact triangle in D′ .

A morphism between exact functors is, by definition, a graded natural transformation.

A functor, which is isomorphic to an exact functor, can be endowed with a structure of a graded functor so that it becomes an exact functor. Indeed, if F is exact , then using isomorphism μ : F →^{∼} G one constructs the natural isomorphism t_{G} : GT →^{∼} T G , t_{G} = μt_{F} μ^{−1} , which makes G graded. Since any triangle , isomorphic to an exact triangle, is again exact, then G transforms exact triangles into exact ones. The natural transformation μ becomes a graded transformation of exact functors.

Let F : D ⟶ D′ be a functor. Suppose we fix a class C of objects in D and for any object X ∈ C some object X′ isomorphic to F X in D′ . If we additionally fix for any X ∈ C an isomorphism F X →^{∼} X′ then there exists a new functor G : D → D′ , which is isomorphic to F and such that

GX = F X, for X ∉ C,
GX = X′, for X ∈ C,  (1)

with the evident action on morphisms.

We shall frequently use this simple fact in the sequel.

Proposition 1.1 i) Let F : D ⟶ D′ be a graded functor between graded categories, G : D′ ⟶ D its left adjoint, so that the natural transformations are given:

id_{D′} ⟶^{α} F ∘ G, G ∘ F ⟶^{β} id_{D}.  (2)

Then G can be canonically endowed with the structure of a graded functor, such that (2) become morphisms of graded functors,

ii) if, in addition, F is an exact functor between triangulated categories, then G also becomes an exact functor. Proof. Let us make G graded.

By the adjointness of G and F and since T_{D} and T_{D′} are equivalences we have the following sequence of bifunctorial isomorphisms:

Hom(GT X , Y) ≅ Hom(T X , F Y) ≅ Hom(X , T^{−1}F Y) ≅
Hom(X , F T^{−1}Y) ≅ Hom(GX , T^{−1}Y) ≅ Hom(T GX , Y)  (3)

for any X ∈ D′, Y ∈ D .

By the well known Brown lemma [Br] this gives a functorial isomorphism:

t_{G} : GT ⟶ T G.

Taking Y = T GX in (3) and tracking carefully the preimage in Hom(GT X , T GX) of id_{T GX} in Hom(T GX , T GX) under the chain of isomorphisms in (3) one obtains a formula for t_{G} . It is, in fact, canonically given as the composition of the following sequence of natural transformations:

GT ⟶^{GT α} GT F G ⟶^{Gt_{F}^{−1}G} GF T G ⟶^{βT G} T G.  (4)

Here we use morphisms α and β from (2) and the grading isomorphism t_{F} for F :

t_{F} : F T ⟶ T F.

To show that, say, α is an isomorphism of graded functors is equivalent to prove that the diagram

T  ⟶^{T α}  T F G  ⟶^{t_{F} G}  F T G
[figure: αT ↓; αT F G; αF T G; ↑; F βT G]
F GT ⟶^{F GT α} F GT F G ⟶^{F Gt_{F} G} F GF T G,

being considered without dotted arrows, is commutative. One can split it by dotted arrows into two commutative squares and the loop, the latter being commutative due to the fact that for adjoint functors the composition

F ⟶^{αF} F GF ⟶^{F β} F

equals id_{F} .

Notice that the inverse morphism to (4) is given by the composition

T G ⟶^{T GT^{−1}αT} T GT^{−1}F GT ⟶^{T GT^{−1}t_{F} T^{−1}GT} T GF T^{−1}GT ⟶^{T βT^{−1}GT} GT.  (5)

That can be found in the same way as (4) by putting Y = GT X in (3).

One needs a lot of commutative diagrams to prove directly, without use of (3), that (4) and (5) are mutually inverse.

ii) [BK] Let A →^{α} B → C → T A be an exact triangle in D′ . We have to show that G transforms this exact triangle into an exact one.

Let us insert the morphism G(α) : GA → GB into an exact triangle:

GA → GB → Z → T GA

Applying functor F to it we obtain an exact triangle:

F GA → F GB → F Z → T F GA

(we use henceforth with no mention the commutation isomorphisms like T F →^{∼} F T ).

By means of id→F G we have a commutative diagram:

A ⟶ B ⟶ C ⟶ T A
↓ ↓ ↓
F GA ⟶ F GB ⟶ F Z ⟶ T F GA

By axioms of triangulated categories there exists a morphism μ : C → F Z that completes this commutative diagram. By adjunction we obtain a morphism ν : GC → Z that makes commutative the following diagram:

GA ⟶ GB ⟶ GC ⟶ T GA
id↓  id↓  ≀↓ν  id↓
GA ⟶ GB ⟶ Z ⟶ T GA

Therefore the lower triangle is exact.

If F is a graded autoequivalence in a graded category, then the adjoint functor is its quasi-inverse.

We may consider a category with object being graded (or respectively triangulated) categories and morphisms being isomorphic classes of graded (respectively exact) equivalences. The proposition ensures that this category is a groupoid. In particular the set of isomorphism classes of graded autoequivalences in a graded category or of exact autoequivalences in a triangulated category is a group.

Now we outline main properties of the Serre functor. Its abstract definition was introduced in [BK].

Definition 1.2 Let D be a k -linear category with finite–dimensional Hom′s . A covariant additive functor S : D → D is called a Serre functor if it is a category equivalence and there are given bi–functorial isomorphisms

φ_{A,B} : Hom_{D}(A , B) ⟶^{∼} Hom_{D}(B , SA)^{∗}

for any A, B ∈ D , with the property that the following diagram is commutative:

Hom_{D}(A , B)  ⟶^{φ_{A,B}} Hom_{D}(B , SA)^{∗}
≀↓  ≀↑
Hom_{D}(SA , SB) ⟶^{φ_{SA,SB}} Hom_{D}(SB , S^{2}A)^{∗}

The vertical isomorphisms in this diagram are induced by S .

Proposition 1.3 Any autoequivalence Φ : D ⟶ D commutes with a Serre functor, i.e. there exists a natural graded isomorphism of functors Φ ∘ S →^{∼} S ∘ Φ .

Proof. For any couple of objects A, B in D we have a system of natural isomorphisms:

Hom(ΦA , ΦSB) ≅ Hom(A , SB) ≅ Hom(B , A)^{∗} ≅ Hom(ΦB , ΦA)^{∗} ≅
≅ Hom(ΦA , SΦB)  (6)

Since Φ is an equivalence the essential image of Φ covers the whole D , i.e. up to isomorphism any object can be presented as ΦA for some A . This means that (6) gives isomorphism of the contravariant functors represented by objects ΦSB and SΦB . By the Brown lemma [Br] morphisms between representable functors are in one-one correspondence with those between the representation objects. This gives isomorphism

ΦSB →^{∼} SΦB,

which is, in fact, natural with respect to B .

Proposition 1.4 i) Any Serre functor in a graded category is graded;

ii) A Serre functor in a triangulated category is exact. Proof. i) follows from previous proposition.

ii) the fact that a Serre functor takes exact triangles into exact ones is proved in [BK].

Proposition 1.5 [BK] Any two Serre functors are connected by a canonical graded functorial isomorphism, which commutes with the bifunctorial isomorphisms ϕ_{A,B} in the definition of Serre functor. Proof. Let S and S′ be two Serre functors in a category D . Then for any object A in D we have natural isomorphisms:

Hom(A , A) ≅ Hom(A , SA)^{∗} ≅ Hom(SA , S′A)

Taking the image of the identity morphism id_{A} with respect to this identification we obtain a morphism SA → S′A , which, in fact, gives a graded functorial isomorphism S →^{∼} S′ , which commutes with ϕ_{A,B} .

Thus, a Serre functor in a category D , if it exists, is unique up to a graded natural isomorphism. By definition it is intrinsically related to the structure of the category. We shall use this later to reconstruct a variety from its derived category and to find the group of exact autoequivalences for algebraic varieties with ample either canonical or anticanonical sheaf.

2 Reconstruction of a variety from the derived category of coherent sheaves.

In this chapter we show that a variety X can be uniquely reconstructed from the derived category of coherent sheaves on it, provided X is smooth and has ample either canonical or anticanonical sheaf. From the category we need only its grading, i.e. fixation of the translation functor.

Roughly, the reconstruction proceeds as follows. First, by means of the Serre functor we distinguish the skyscraper sheaves of closed points in the variety. Then we find the invertible sheaves and use them to define the Zarisky topology and the structure sheaf for the variety.

Let D be a k–linear category. Denote by S_{D} the Serre functor in D (for the case it exists).

Let X be a smooth algebraic variety, n = dimX , D = D_{coh}^{b}(X) the derived category of coherent sheaves on X and ω_{X} the canonical sheaf. Then the functor

(·) ⊗ ω_{X}[n]  (7)

is the Serre functor in D , in view of the Serre–Grothendieck duality:

Ext^{i}(F , G) = Ext^{n−i}(G , F ⊗ ω_{X})^{∗}

for any couple F, G coherent sheaves on X ([S], [G]).

For derived categories the translation functor we consider is always the usual shift of grading.

For a closed point x ∈ X we denote by k(x) the residue field of this point. We use the standard notations for iterated action of the translation functor on an

object P :

P [i] := T^{i}P, i ∈ Z,

and for the composition of the functors Hom and T :

Hom^{i}(P , Q) = Ext^{i}(P, Q) := Hom(P , Q[i]).

Definition 2.1 An object P ∈ D is called point object of codimension s , if

\begin{matrix}i) & S_{D}(P) ≃ P [s], \\ ii) & Hom^{<0}(P , P) = 0, \\ iii) & Hom^{0}(P , P) = k(P). \end{matrix}

with k(P) being a field (which is automatically a finite extension of the basic field k ).

Proposition 2.2 Let X be a smooth algebraic variety of dimension n with the ample canonical or anticanonical sheaf. Then an object P ∈ D_{coh}^{b}(X) is a point object, iff P ≅ O_{x}[r] , r ∈ Z , is isomorphic (up to translation) to the skyscraper sheaf of a closed point x ∈ X . Remark. Since X has an ample invertible sheaf it is projective. Proof. Any skyscraper sheaf of a closed point obviously satisfies properties of a point object of the same codimension as the dimension of the variety.

Suppose now that for some object P ∈ D_{coh}^{b}(X) properties i)–iii) of definition 2.1 are verified.

Let H^{i} are cohomology sheaves of P . It immediately follows from i) that s = n and H^{i} ⊗ ω_{X} = H^{i} . Since ω_{X} is either ample or antiample sheaf, we conclude that H^{i} are finite length sheaves, i.e. their support are isolated points. Sheaves with the support in different points are homologically orthogonal, therefore any such object decomposes into direct some of those having the support of all cohomology sheaves in a single point. By iii) the object P is indecomposable. Now consider the spectral sequence, which calculates Hom^{m}(P , P) by Ext^{i}(H^{j} , H^{k}) :

E_{2}^{p,q} = ⨁_{k−j=q} Ext^{p}(H^{j} , H^{k}) ⟹ Hom^{p+q}(P , P).

Let us mention that for any two finite length sheaves having the same single point as their support, there exists a non–trivial homomorphism from one to the other, which sends generators of the first one to the socle of the second.

Considering Hom^{m}(H^{j} , H^{k}) with minimal k − j , we observe that this non– trivial space survives at E_{∞} , hence by ii) k − j = 0 . That means that all but one cohomology sheaves are trivial. Moreover, iii) implies that this sheaf is a skyscraper. This concludes the proof.

Now having the skyscrapers we are able to reconstruct the invertible sheaves.

Definition 2.3 An object L ∈ D is called invertible if for any point object P ∈ D there exists s ∈ Z such that

i) Hom^{s}(L , P) = k(P),
ii) Hom^{i}(L , P) = 0,  for i ≠ s.

Proposition 2.4 Let X be a smooth irreducible algebraic variety. Assume that all point objects have the form O_{x}[s] for some x ∈ X, s ∈ Z . Then an object L ∈ D is invertible, iff L ≅ L[t] for some invertible sheaf L on X , t ∈ Z .

Proof. For an invertible sheaf L we have:

Hom(L , O_{x}) = k(x), Ext^{i}(L , O_{x}) = 0, if i ≠ 0.

Therefore, if L = L[s] , then it is an invertible object.

Now let H^{i} are the cohomology sheaves for an invertible object L . Consider the spectral sequence that calculates Hom^{.}(L , O_{x}) for a point x ∈ X by means of Hom^{i}(H^{j} , O_{x}) :

E_{2}^{p,q} = Hom^{p}(H^{q} , O_{x}) ⟹ Ext^{p−q}(L , O_{x}).

Let H^{q_{0}} be the nontrivial cohomology sheaf with maximal index. Then for any closed point x ∈ X from the support of H^{q_{0}} Hom(H^{q_{0}} , O_{x}) ≠ 0 . But both Hom(H^{q_{0}} , O_{x}) and Ext^{1}(H^{q_{0}} , O_{x}) are intact by differential of the spectral sequence. Therefore, by definition of an invertible object we obtain that for any point x from the support of H^{q_{0}}

\begin{matrix}a) & Hom(H^{q_{0}} , O_{x}) = k(x), \\ b) & Ext^{1}(H^{q_{0}} , O_{x}) = 0. \end{matrix}

Since X is smooth and irreducible it follows from b) that the H^{q_{0}} is locally free on X . From a) one deduces that it is invertible.

This implies that Ext^{i}(H^{q_{0}} , O_{x}) = 0 for i > 0 and Hom(H^{q_{0}−1}, O_{x}) are intact by differentials of the spectral sequence. This means that Hom(H^{q_{0}−1} , O_{x}) = 0 , for any x ∈ X , i.e. H^{q_{0}−1} = 0 . Repeating this argument for H^{q} with smaller q , we easily see that all H^{q} , except q = q_{0} , are trivial. This proves the proposition.

Now we are ready to prove the reconstruction theorem. Invertible sheaves help us to ‘glue’ points together.

Theorem 2.5 Let X be a smooth irreducible projective variety with ample canonical or anticanonical sheaf. If D = D_{coh}^{b}(X) is equivalent as a graded category to D_{coh}^{b}(X′) for some other smooth algebraic variety X′ , then X is isomorphic to X′ .

This theorem is stronger than just a reconstruction for a variety with ample canonical or anticanonical sheaf from its derived category. Let us mention that since X′ might not have ample canonical or anticanonical sheaf, the situation is not symmetric with respect to X and X′ .

We divide the proof in several steps, so that the reconstruction procedure will be transparent. Proof. During the proof while saying that two isomorphism classes of objects, one in D_{coh}^{b}(X) and the other in D_{coh}^{b}(X′) , are equal we mean that the former is taken to the latter by the primary equivalence D_{coh}^{b}(X) ⟶^{∼} D_{coh}^{b}(X′) .

Step 1. Denote P_{D} the set of isomorphism classes of the point objects in D , P_{X} the set of isomorphism classes of objects in D_{coh}^{b}(X)

P_{X} := {O_{x}[k] | x ∈ X, k ∈ Z}.

By proposition 2.2 P_{D} ≅ P_{X} . Obviously, P_{X′} ⊂ P_{D} . Suppose that there is an object P ⊂ P_{D} , which is not contained in P_{X′} . Since P_{D} ≅ P_{X} , any two objects in P_{D} either are homologically mutually orthogonal or belong to a common orbit with respect to the translation functor. It follows that P ∈ D_{coh}^{b}(X′) is orthogonal to any skyscraper sheaf O_{x′}, x′ ∈ X′ . Hence P is zero. Therefore, P_{X′} = P_{D} = P_{X} .

Step 2. Denote by L_{D} the set of isomorphism classes of invertible objects in D , L_{X} the set of isomorphism classes of objects in D_{coh}^{b}(X)

L_{X} := {L[k] | L being an invertible sheaf on X, k ∈ Z}.

By step 1 both varieties X and X′ satisfy the assumptions of proposition 2.4 . It follows that L_{X} = L_{D} = L_{X′} .

Step 3. Let us fix some invertible object L_{0} in D , which is an invertible sheaf on X . By step 2 L_{0} can be regarded, up to translation, as an invertible sheaf on X′ . Moreover, changing, if necessary, the equivalence D_{coh}^{b}(X) ≃ D_{coh}^{b}(X′) , by the translation functor, we can assume that L_{0} , regarded as an object on X′ , is a genuine invertible sheaf.( Formally speaking L_{0} is taken by the equivalence D_{coh}^{b}(X) ⟶^{∼} D_{coh}^{b}(X′) to an object which is isomorphic to an invertible sheaf on X′ . But as was explained in chapter 1 (formula (1)) we can adjust this equivalence so that it takes L_{0} into the invertible sheaf on X′ .)

Obviously, by step 1 the set p_{D} ⊂ P_{D}

p_{D} := {P ∈ P_{D} | Hom(L , P) = k(P)}

coincides with both sets p_{X} = {O_{x}, x ∈ X} and p_{X′} = {O_{x′}, x′ ∈ X′} . This gives us a pointwise identification of X and X′ .

Step 4. Let now l_{X} (resp., l_{X′} ) be the subset in L_{D} of isomorphism classes of invertible sheaves on X (resp., on X′ ).

They can be recognized from the graded category structure in D as follows:

l_{X′} = l_{X} = l_{D} := {L ∈ L_{D} | Hom(L , P) = k(P) for any P ∈ p_{D}}.

For α ∈ Hom(L_{1} , L_{2}) , where L_{1}, L_{2} ∈ l_{D} , and P ∈ p_{D} , denote by α_{P}^{∗} the induced morphism:

α_{P}^{∗} : Hom(L_{2} , P) ⟶ Hom(L_{1} , P),

and by U_{α} the subset of those objects P in p_{D} for which α_{P}^{∗} ≠ 0 . By [Il] any algebraic variety has an ample system of invertible sheaves. This means that U_{α} , where α runs over all elements in Hom(L_{1} , L_{2}) and L_{1} and L_{2} runs over all elements in l_{D} , give a base for the Zariski topologies on both X and X′ . It follows that the topologies on X and X′ coincide.

Step 5. Since codimensions of all point objects are equal to the dimensions of X and of X′ , we have dimX = dimX′ . Then, formula (7) for the Serre functor shows that the operations of twisting by the canonical sheaf on X and on X′ induce equal transformations on the set l_{D} .

Let L_{i} = F^{i}L_{0}[−ni] . Then {L_{i}} is the orbit of L_{0} with respect to twisting by the canonical sheaf on X . Changing, if necessary, the equivalence D_{coh}^{b}(X) ⟶^{∼} D_{coh}^{b}(X′) we can assume that {L_{i}} is the orbit of L_{0} with respect to twisting by the canonical sheaf on X′ too.

Since the canonical sheaf ω_{X} is either ample or antiample, the set of all U_{α} , where α runs over all elements in Hom(L_{i} , L_{j}), i, j ∈ Z , is the base of the Zariski topology on X , hence, by step 4, on X′ . This means that canonical sheaf on X′ is also ample or, respectively, antiample (see [Il]).

For all pairs (i, j) there are natural isomorphisms:

Hom(L_{i} , L_{j}) ≅ Hom(S^{i}L_{0}[−ni] , S^{j}L_{0}[−nj]) ≅
≅ Hom(L_{0} , S^{j−i}L_{0}[−n(j − i)]) ≅ Hom(L_{0} , L_{j−i}).

They induce a ring structure in the graded algebra A over k with graded components

A_{i} = Hom(L_{0} , L_{i}).

This algebra , being defined intrinsicly by the graded category structure, is isomorphic to the coordinate algebra B of the canonical sheaf for X , i.e. algebra with graded components:

B_{i} = Hom_{X}(O_{X}, ω_{X}^{⊗i}).

Indeed, L_{i} = L_{0} ⊗ ω_{X}^{⊗i} , the isomorphism being given by tensoring by L_{0} . It is a ring homomorphism, because the functor of tensoring by L_{0} commutes with the Serre functor by proposition 1.3 .

The same is true for the coordinate algebra B′ of the canonical sheaf for X′ . Eventually, we obtain isomorphism B →^{∼} B′ of the canonical algebras on X and X′ . Since the canonical sheaves on both X and X′ are ample ( or antiample), both varieties can be obtained by projectivization from the canonical algebras:

X →^{∼} ProjB →^{∼} ProjB′ →^{∼} X′

This gives a biregular isomorphism between X and X′ as algebraic varieties. This finishes the proof.

3 Group of exact autoequivalences.

It was explained in chapter 1 that the set of isomorphism classes of exact autoequivalences in a triangulated category D is a group. We denote this group by AutD .

The problem of reconstructing of a variety from its derived category is closely related to the problem of computing the group of exact autoequivalences for D_{coh}^{b}(X) . For ample canonical or anticanonical sheaf we have the following

Theorem 3.1 Let X be a smooth irreducible 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. Proof. Assume for definiteness that the canonical sheaf is ample. Choose an autoequivalence F . Since the class of invertible objects is defined intrinsicly with respect to the graded structure of the category, it is preserved by any autoequivalence. Moreover, the set of isomorphism classes of invertible objects is transitive with respect to the action of the subgroup AutD_{coh}^{b}(X) generated by translations and twists. Indeed, by propositions 2.2 and 2.4 all invertible objects in D_{coh}^{b}(X) are invertible sheaves up to translations. Any invertible sheaf can be obtained from the trivial sheaf O by applying the functor of tensoring with this invertible sheaf. Therefore, using twists with invertible sheaves and translations we can assume that our functor F takes O to O . It follows that F takes any tensor power ω_{X}^{⊗i} of the canonical sheaf into itself, because by proposition 1.3 it commutes with the Serre functor.

Therefore, our functor induces an automorphism of the graded coordinate algebra A of the canonical sheaf , i.e. algebra with graded components:

A_{i} = Hom(O , ω_{X}^{⊗i}) = H^{0}(ω_{X}^{⊗i})

Any graded automorphism of the canonical algebra induces an automorphism of the variety. Adjusting our functor F by an autoequivalence induced by an automorphism of the variety we can assume that the automorphism of the canonical algebra induces the trivial automorphism of the variety.

Such an automorphism is actually a scaling, i.e. it takes an element a ∈ H^{0}(ω_{X}^{⊗i}) to λ^{i}a , for some fixed scalar λ . Indeed, the graded ideal generated by any element a ∈ H^{0}(ω_{X}^{⊗i}) is stable with respect to the automorphism. It follows that a is multiplied by a scalar. Then the linear operator in the graded component H^{0}(ω_{X}^{⊗i}) induced by the automorphism should be scalar , say λ_{i} . Since our automorphism is algebra automorphism, it follows that λ_{i} = λ^{i} , for λ = λ_{1} (in case H^{0}(ω_{X}) = 0 , i.e. when λ_{1} is not defined, we may substitute in this reasoning the Serre functor by a sufficient j -th power of it such that H^{0}(ω_{X}^{⊗j}) ≠ 0 , and respectively the canonical algebra by the corresponding Veronese subalgebra).

To kill the scaling of the canonical algebra we substitute functor F by an isomorphic one. For this we take the subclass C of objects in D consisting of powers of the canonical sheaf:

C = {ω_{X}^{⊗i}}, i ∈ Z.

As in chapter 1 for any object C in C we need to choose an isomorphism of its image with some other object C′ . Our functor preserves all objects from C . We choose C′ = C for any C in C and the non–trivial isomorphism: if C = ω_{X}^{⊗i} , then the isomorphism is λ^{−i} · id_{C} .

Then the new functor constructed by formula (1) induces the trivial automorphism of the canonical ring.

Thus we have a functor, which takes the trivial invertible sheaf and any power of the canonical sheaf to themselves and preserves homomorphisms between all these sheaves. Let us show that such a functor is isomorphic to the identity functor.

First, our functor takes pure sheaves to objects, isomorphic to pure sheaves, because such objects can be characterized as the objects G in D_{coh}^{b}(X) having trivial Hom^{k}(ω_{X}^{⊗i}, G) , for k ≠ 0 and for sufficiently negative i . Again we can substitute our functor by an isomorphic one, which takes sheaves to pure sheaves. By Serre theorem [S] the abelian category of pure sheaves is equivalent to the category of graded finitely generated modules over the canonical algebra A modulo the subcategory of finite dimensional modules. The equivalence takes a sheaf G into a module M(G) with graded components:

M_{i}(G) = Hom(ω_{X}^{⊗−i} , G)

Our functor F gives isomorphisms:

Hom(ω_{X}^{⊗−i} , G) ⟶^{∼} Hom(F (ω_{X}^{⊗−i}) , F (G)) ≅ Hom(ω_{X}^{⊗−i} , F (G))

Since F induces trivial action on the canonical algebra these isomorphisms form an isomorphism of A -modules:

M(G) →^{∼} M(F (G)).

It is natural with respect to G . Hence we obtain an isomorphism of functors

M →^{∼} M ∘ F .

Since modulo the subcategory of finite dimensional modules M is an equivalence, we have a functorial isomorphism id →^{∼} F on the subcategory of coherent sheaves.

Our system of objects {ω_{X}^{⊗i}} enjoys nice properties with respect to the abelian category of coherent sheaves on X , which allow to extend the natural transformation id → F , from the core of the t-structure to a natural isomorphism in the whole derived category. It was done in Proposition A.3 of the Appendix.

This finishes the proof of the theorem.

In the hypothesis of Theorem 3.1 the group AutD_{coh}^{b}(X) is the semi-direct product of its subgroups G_{1} = PicX ⊕ Z and G_{2} = AutX , Z being generated by the translation functor:

AutD_{coh}^{b}(X) ≅ AutX ⋉ (PicX ⊕ Z)

Indeed, in course of the proof of the theorem we in fact showed that any element from AutD_{coh}^{b}(X) could be decomposed as g = g_{1}g_{2} with g_{1} ∈ G_{1} and g_{2} ∈ G_{2} . The subgroups G_{1} and G_{2} meet trivially in G , because the elements from the latter take the structure sheaf O to itself, while those from the former do not. Group G_{1} is obviously preserved by conjugation by elements from G_{1} and G_{2} , hence normal in G .

Appendix.

This appendix is devoted to describing conditions, under which one can extend to the whole category a natural isomorphism between the identity functor and an exact autoequivalence in the bounded derived category D_{coh}^{b}(A) , provided one has such an isomorphism in an abelian category A (or even in a smaller subcategory, see the proposition below).

To break our way through technical details we need a sequence of objects in the abelian category with some remarkable properties. For the case when the sequence consists of powers of an invertible sheaf these properties are resulted from ampleness of this sheaf. For this reason we postulate them under the name of ampleness.

Definition A.1 Let A be an abelian category. We call a sequence of objects {P_{i}}, i ∈ Z_{≤0} , ample if for every object X ∈ A , there exists N such that for all i < N the following conditions hold:

a) the canonical morphism Hom(P_{i} , X) ⊗ P_{i} ⟶ X is surjective,

b) Ext^{j}(P_{i} , X) = 0 for any j ≠ 0 , c) Hom(X , P_{i}) = 0 . Denote by D^{b}(A) the bounded derived category of A . Let us consider A as

a full subcategory j : A ↪ D^{b}(A) in D^{b}(A) in the usual way. We also consider a full subcategory q : C ↪ D^{b}(A) with ObC = {P_{i}}_{i∈Z_{≤0}} . We shall show that if there exists an exact autoequivalence F : D^{b}(A) ⟶ D^{b}(A) and an isomorphism of its restriction to C with the identity functor id_{C} , then this isomorphism can be uniquely extended to an isomorphism of F with the identity functor id_{D^{b}(A)} in the whole D^{b}(A) .

The idea is in reducing the number of cohomology for an object by killing the highest one by means of a surjective morphism from ⊕P_{i} for sufficiently negative i .

For the proof we shall repeatedly use the following lemma (see [BBD]).

Lemma A.2 Let g be a morphism from Y to Y′ and suppose that these objects are included into the following two exact triangles:

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1]
[figure: f; ↓g; h; f[1]]
X′ ⟶^{u′} Y′ ⟶^{v′} Z′ ⟶^{w′} X′[1]

If v′gu = 0 , then there exist morphisms f : X → X′ and h : Z → Z′ such that the triple (f, g, h) is a morphism of triangles.

If, in addition, Hom(X[1] , Z′) = 0 , then the morphisms f and h , making commutative the first and, respectively, the second square of the diagram, are unique.

Proposition A.3 Let A be an abelian category possessing an ample sequence {P_{i}} and let F : D^{b}(A) ⟶ D^{b}(A) be an exact autoequivalence. Suppose there exists an isomorphism f : q ⟶^{∼} F |_{C} (where q : C ↪ D^{b}(A) is the natural embedding). Then this isomorphism can be uniquely extended to an isomorphism id ⟶^{∼} F in the whole D^{b}(A) . Proof. Note that X ∈ D^{b}(A) is isomorphic to an object in A iff Hom^{j}(P_{i}, X) = 0 for j ≠ 0 and i ≪ 0 , in view of the condition b) from Definition A.1.

This allows us to ”extract” the abelian subcategory A from D^{b}(A) by means of the sequence {P_{i}} . Then using surjective coverings Hom(P_{i} , X) ⊗ P_{i} ⟶ X by standard techniques from the theory of abelian categories one can extend f to an isomorphism (which we denote by the same letter) f : j ⟶^{∼} F |_{A} , where j stands for the natural embedding j : A ↪ D^{b}(A) . We skip details of this part of the proof, because we don’t need it in the main body of the paper.

Let us define f_{X[n]} : X[n] ⟶ F (X[n]) ≅ F (X)[n] for X ∈ A by

f_{X[n]} = f_{X}[n].

It is not difficult to show that for any X and Y in A and for any u ∈ Ext^{k}(X , Y) the diagram

\begin{matrix}X & ⟶^{u} & Y [k] \\ f_{X} ↓ & & ↓f_{Y} [k] \\ F (X) & ⟶^{F (u)} & F (Y)[k]. \end{matrix}  (8)

is commutative. Indeed, since any element u ∈ Ext^{k}(X , Y) can be represented as the Yoneda composition u = u_{1} ⋯ u_{k} of elements u_{i} ∈ Ext^{1}(Z_{i} , Z_{i+1}) for some objects Z_{i} , with Z_{1} = X, Z_{k+1} = Y , we can restrict ourselves to the case u ∈ Ext^{1}(X , Y) . Consider the following diagram:

Y ⟶ Z ⟶^{p} X ⟶^{u} Y [1]
f_{Y} ↓  ↓f_{Z}  [figure: h]  ↓f_{Y} [1]
F (Y) ⟶ F (Z) ⟶^{F (p)} F (X) ⟶^{F (u)} F (Y)[1]

By an axiom of triangulated categories there exists a morphism h : X → F (X) such that (f_{Y} , f_{Z}, h) is a morphism of triangles. On the other hand, since Hom(Y [1] , F (X)) = 0 , by the lemma above h is a unique morphism such that F (p) ∘ f_{Z} = h ∘ p . As F (p)f_{Z} = f_{X}p , we conclude that h = f_{X} . This implies commutativity of the diagram (8) for k = 1 .

We shall prove by induction over n the following statement. Consider the full subcategory j_{n} : D_{n} ↪ D^{b}(A) in D^{b}(A) generated by objects having nontrivial cohomology in a (non-fixed) segment of length n . Then there is a unique extention of f to a natural functorial isomorphism f_{n} : j_{n} ⟶ F |_{D_{n}} .

Above we have completed the first, n = 1 , step of the induction.

Now take the step n = a, a ≥ 1 , for granted. Let X be an object in D_{a+1} and suppose, for definiteness, that its cohomology H^{p}(X) are nontrivial only for p ∈ [−a, 0] . Take P_{i} from the given ample sequence with sufficiently negative i such that

a) Hom^{j}(P_{i} , H^{p}(X)) = 0 for all p and for j ≠ 0,
b) there exists a surjective morphism u : P_{i}^{⊕k} ⟶ H^{0}(X),
c) Hom(H^{0}(X) , P_{i}) = 0.

(9)

Note that in view of condition a) and the standard spectral sequence Hom(P_{i} , X) →^{∼} Hom(P_{i} , H^{0}(X)) . This means that we can find a morphism v : P_{i}^{⊕k} ⟶ X such that the composition of v with the canonical morphism X ⟶ H^{0}(X) coincides with u . Consider an exact triangle:

Y [−1] ⟶ P_{i}^{⊕k} ⟶^{v} X ⟶ Y.

Denote by f_{i} the morphism f_{X} for X = P_{i}^{⊕k} . Since Y belongs to D_{a} by the induction hypothesis, the isomorphism f_{Y} is already defined and the diagram:

\begin{matrix}P_{i}^{⊕k} & ⟶^{v} & X & ⟶ & Y & ⟶ & P_{i}^{⊕k}[1] \\ ↓f_{i} & & [figure: f_{X}] & & ↓f_{Y} & & ↓f_{i}[1] \\ F (P_{i}^{⊕k}) & ⟶^{F (v)} & F (X) & ⟶ & F (Y) & ⟶ & F (P_{i}^{⊕k})[1] \end{matrix}  (10)

is commutative.

Further, we have the following sequence of isomorphisms:

Hom(X , F (P_{i}^{⊕k})) ≅ Hom(X , P_{i}^{⊕k}) ≅ Hom(H^{0}(X) , P_{i}^{⊕k}) = 0.

Hence, applying lemma A.2 to g equal f_{Y} , we obtain a unique morphism f_{X} : X ⟶ F (X) that preserves commutativity of the above diagram.

It is clear from the definition that f_{X} is an isomorphism, if so are f_{i} and f_{Y} . For the future we need to show that f_{X} does not depend on the choice for i and u . Suppose we are given two surjective morphisms u_{1} : P_{i_{1}}^{⊕k_{1}} ⟶ H^{0}(X) and u_{2} : P_{i_{2}}^{⊕k_{2}} ⟶ H^{0}(X) , where i_{1} and i_{2} are sufficiently negative to satisfy conditions a), b) and c). Then we can find sufficiently negative j and surjective morphisms w_{1}, w_{2} such that the following diagram commutes:

P_{j}^{⊕l} ⟶^{w_{2}} P_{i_{2}}^{⊕k_{2}}
↓w_{1}  ↓u_{2}
P_{i_{1}}^{⊕k_{1}} ⟶^{u_{1}} H^{0}(X).

Denote by v_{1} : P_{i_{1}}^{⊕k_{1}} ⟶ X, v_{2} : P_{i_{2}}^{⊕k_{2}} ⟶ X the morphisms corresponding to u_{1} and u_{2} . Since Hom(P_{j} , X) →^{∼} Hom(P_{j} , H^{0}(X)) , we have v_{2}w_{2} = v_{1}w_{1} .

There is a morphism ϕ : Y_{j} ⟶ Y_{i_{1}} such that the triple (w_{1}, id, ϕ) is a morphism of exact triangles:

P_{j}^{⊕l} ⟶^{v_{1}∘w_{1}} X ⟶^{y} Y_{j} ⟶ P_{j}^{⊕l}[1]
w_{1}↓  ↓id  ↓ϕ  ↓w_{1}[1]
P_{i_{1}}^{⊕k_{1}}  ⟶^{v_{1}} X ⟶^{y_{1}} Y_{i_{1}} ⟶ P_{i_{1}}^{⊕k_{1}}[1],

i.e. ϕy = y_{1} .

Since Y_{j} and Y_{i_{1}} have cohomology in the segment [−a, −1] , by the induction hypothesis, the following square is commutative:

Y_{j}  ⟶^{ϕ} Y_{i_{1}}
f_{Y_{j}} ↓  ↓f_{Y_{i1}}
F (Y_{j}) ⟶^{F (ϕ)} F (Y_{i_{1}}).

Denote by f_{X}^{j}, f_{X}^{i_{1}}, f_{X}^{i_{2}} the unique morphisms constructed as above to make commutative the diagram (10) for v equal respectively v = v_{1}w_{1}, v = v_{1}, v = v_{2} . Further, we have:

F (y_{1})f_{X}^{j} = F (ϕy)f_{X}^{j} = F (ϕ)F (y)f_{X}^{j} = F (ϕ)F_{Y_{j}}y = f_{Y_{i_{1}}} ϕy = f_{Y_{i_{1}}} y_{1}.

It follows that f_{X}^{j} = f_{X}^{i_{1}} . Analogously, since v_{1}w_{1} = v_{2}w_{2} we have f_{X}^{j} = f_{X}^{i_{2}} . Therefore the morphism f_{X} does not depend on the choice of i and of the morphism u : P_{i}^{⊕k} ⟶ H^{0}(X) .

By means of the translation functor we obtain in the obvious way the only possible extension of f_{a} to D_{a+1} . Let us prove that it is indeed a natural transformation from j_{a+1} to F |_{D_{a+1}} , i.e. that for any morphism ϕ : X ⟶ Y , X, Y being in D_{a+1} , the following diagram commutes

\begin{matrix}X & ⟶^{ϕ} & Y \\ f_{X} ↓ & & ↓f_{Y} \\ F (X) & ⟶^{F (ϕ)} & F (Y). \end{matrix}  (11)

We shall reduce the problem to the case when both X and Y are in D_{a} .

There are two working posibilities that we shall utilize for this.

Case 1. Suppose that the upper bound, say 0 (without loss of generality), of cohomology for X is greater than that for Y . Take a surjective morphism u : P_{i}^{⊕k} ⟶ H^{0}(X) satisfying a), b), c) and construct the morphism v : P_{i}^{⊕k} ⟶ X related to u as above. We have an exact triangle:

P_{i}^{⊕k} ⟶^{v_{1}} X ⟶^{α} Z ⟶ P_{i}^{⊕k}[1].

If we take i sufficiently negative, then Hom(P_{i}^{⊕k} , Y) = 0 . Applying the functor Hom(− , Y) to this triangle we found that there exists a morphism ψ : Z ⟶ Y such that ϕ = ψα . We know that f_{X} , defined above, satisfies the equation

F (α)f_{X} = f_{Z}α.

If we assume that

F (ψ)f_{Z} = f_{Y} ψ,

then

F (ϕ)f_{X} = F (ψ)F (α)f_{X} = F (ψ)f_{Z}α = f_{Y} ψα = f_{Y} ϕ.

This means that for this case in verifying commutativity of (11) we can substitute X by an object Z such that the upper bound of its cohomology is less by one than that for X . Moreover, one can easily see that if X belongs to D_{k} , with k > 1 , then Z does to D_{k−1} , and if it is in D_{1} then so is Z .

Case 2. Suppose now that the upper bound , say 0 (again without loss of generality), of cohomology for Y is greater than or equal to that for X . Take a surjective morphism u : P_{i}^{⊕k} ⟶ H^{0}(Y) with i satisfying a), b), c) (with Y instead of X ) and construct a morphism v : P_{i}^{⊕k} ⟶ Y related to u . Consider an exact triangle

P_{i}^{⊕k} ⟶^{v} Y ⟶^{β} W ⟶ P_{i}^{⊕k}.

By ψ denote the composition β ∘ ϕ .

If we assume that

F (ψ)f_{X} = f_{W} ψ,

then, since F (β)f_{Y} = f_{W} β we have:

F (β)(f_{Y} ϕ − F (ϕ)f_{X}) = f_{W} βϕ − f(βϕ)f_{X} = f_{W} ψ − F (ψ)f_{X} = 0.  (12)

We again take i sufficiently negative, so that Hom(X , P_{i}^{⊕k}) = 0 . As F (P_{i}^{⊕k}) is isomorphic to P_{i}^{⊕k} , then Hom(X , F (P_{i}^{⊕k})) = 0 . Applying the functor Hom(X , F (−)) to the above triangle we found that the composition with F (β) gives an inclusion of Hom(X , F (Y)) into Hom(X , F (W)) . It follows from (12) that f_{Y} ϕ = F (ϕ)f_{X} .

Thus in this case in verifying commutativity of (11) we can substitute Y by an object W such that the upper bound of its cohomology is less by one than that for Y . If Y belongs to D_{k}, k > 1 , then W does to D_{k−1} , if Y belongs to D_{1} , then so does W .

Suppose now that X and Y are in D_{a+1}, a > 1 . Depending on which case, 1) or 2), we are in, we can substitute either X or Y by an object lying in D_{a} . Then repeating, if necessary, the procedure we can lower the upper bound of the cohomology of the object to such a point that the other case is applicable. Then we shorten the cohomology segment of the second object and come to the situation when both objects are in D_{a} , i.e. to the induction hypothesis.

At every step of the construction we always made the only possible choice for the morphism f_{X} . This means that the natural transformation with required properties is unique.

This finishes the proof of the proposition.

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Algebra section, Steklov Math. Institute, Vavilova 42, 117333 Moscow, RUSSIA bondal@sovam.com , orlov@class.mi.ras.ru