Twisting Derived Equivalences

We introduce a new method for ``twisting'' relative equivalences of derived categories of sheaves on two spaces over the same base. The first aspect of this is that the derived categories of sheaves on the spaces are twisted. They become derived categories of sheaves on gerbes living over spaces that are locally (on the base) isomorphic to the original spaces. Secondly, this is done in a compatible way so that the equivalence is maintained. We apply this method by proving the conjectures of Donagi and Pantev on dualities between gerbes on genus-one fibrations and comment on other applications to families of higher genus curves. We also include a related conjecture in Mirror Symmetry. This represents a modified version of my May 2006 thesis for the University of Pennsylvania. The only addition is the subsection titled 'Alternative Method'.


Introduction
Categories of sheaves of modules on geometric objects seem to play an important role in algebraic geometry. It is sometimes possible to analyze the algebraic properties of such categories in detail, which in turn sheds light on the nature of the space itself. On the other hand, they also suggest a broader perspective in which a space can vary in "non-geometric" directions, and eventually the notion of a space could be replaced by a category with certain properties. Gabriel showed in [21] that one can recover a Noetherian scheme from its category of coherent sheaves. Alternatively, one can study the derived category of coherent sheaves. This is a less rigid structure, and certainly allows for different spaces to have equivalent derived categories. This "derived equivalence" game started with Mukai's equivalence of dual complex tori [29] and we give a quick summary in what follows. The fundamental question which motivates this work (although we only scratch its surface), is, How does the derived category vary in families?. Although the derived category is certainly known not to glue, in some cases it seems to behave as if it does. That is to say, if one is careful enough, one can prove things that would easily follow if descent for derived categories held. In this thesis, we investigate the special case when the gluing takes place on the level of the abelian category of sheaves (which as a stack can be glued from its restriction to a cover). Rather than directly study these questions for the derived category of a space, we study the variance (or twisting) of pairs of derived categories related by a Fourier-Mukai type equivalence. In practice, we start with a given Fourier-Mukai equivalence between the derived categories of two spaces X and Y . We ask: If the category of sheaves on X is twisted in some way, can the derived category of Y be twisted in a compatible way in order to recover a new equivalence? On the infinitesimal level, this question was addressed by Y. Toda in [32]. A formal analysis to all orders was completed, in a special case, in [4]. A complete picture of dualities for arbitrary gerby deformation-quantizations of abelian schemes has been completed by D. Arinkin, building on the results from his 2002 Thesis [3]. A non-formal version of this same case was carried out by J. Block in [5,6]. Topological versions of twisted T-dualities have been studied by Mathai and Rosenberg in [27,28] and by Bunke, Rumpf, and Schick in [12]. There is also groupoid approach due to these dualities due to C. Daenzer [16]. It is somewhat interesting and strange that the philosophy of Toda from [32] will be helpful to us, even though our twistings are not necessarily deformations, and certainly not formal deformations. According to this philosophy, for every derived equivalence between X and Y , one should be able to first find an algebraic object describing the twistings of the derived category D(X) compatible with the derived equivalence. Next one should find a natural isomorphism with these twistings of D(X) and the analogous twistings of D(Y ). Finally, one should find an equivalence between the categories associated to a pair of compatible twistings. In this thesis, we synthesize two types of twistings. The first are those coming from replacing a space by another space locally isomorphic to it. The second are those coming from replacing the derived category of sheaves on a space with the derived category of sheaves on a gerbe over the space. The most convenient setting therefore becomes the one where X and Y are both fibered over the same space B, and the derived equivalence respects that structure. In this case, our twists of the derived categories correspond to decomposing and then re-gluing the relative stack of the abelian categories of sheaves of the two spaces. We can now state our main theorem (with slight re-wording), to be proven as 5.11. We use here the definitions of Φ-compatible and Φ-dual which can be found as definitions 5.6 and 5.8 respectively. Theorem 1.1 Let X and Y be compact, connected, complex manifolds, mapping to a complex analytic space B, via maps π : X → B and ρ : Y → B, where ρ is flat. Let P be a coherent sheaf, flat over Y , on the fiber product X × B Y , which gives an equivalence of categories Φ : D b c (Y ) → D b c (X), Φ = Rφ with φ(S) = ρ * (P ⊗ π * S).
Then for any Φ-compatible gerbe X over a twisted version of X → B, and any Φ-dual gerbe Y to X, there is an equivalence of categories where Φ = R φ and φ is locally built out of φ.
The motivation for such a theorem begins at least with Dolgachev and Gross [18], who relate the Tate-Shafarevich group of an elliptic fibration to its Brauer group. This naturally leads one to ask if the derived category of a geometrically twisted elliptic fibration (a genus one fibration) is somehow related to a derived category of sheaves on a gerbe over the elliptic fibration. Such a relationship was indeed shown in [15,14]. More general dualities involving gerbes on genus one fibrations were proven in [19]. The conjectures made in [19] form the main geometric motivation for this thesis, although the methods of proof, will be closer to those found in [14]. More recently, these kind of results have been used and expanded upon by other authors, for example see [11], [13], and [31].
After the thesis was submitted, we added an alternative, and in many ways stronger, version of the main theorem which can be found as 5.12. In the last section, we comment on some future, more exotic applications of the ideas we have developed including applications to homological mirror symmetry, and hyperelliptic families in algebraic geometry.

Basic Information
In this thesis, we work with complex manifolds, or complex analytic spaces and we always use the classical (analytic) topology. If S is a sheaf of abelian groups on a topological space X, and U is a cover of X, then we denote byČ U (X, S),Ž U (X, S), andB U (X, S) theČech co-chains, co-cycles, and coboundaries, for the sheaf of abelian groups S computed with respect to the cover U of X. When we are dealing with a ringed space (or stack), by a sheaf (with no qualifications) we always mean a sheaf of modules for the structure sheaf of the space (or stack).
For our purposes, it will always suffice to consider stacks on a complex analytic space X which are defined on the site whose underlying category consists of analytic spaces over X, and whose covering families for a given W → X consist of surjective local isomorphisms {W ′ → W } over X. We will often consider the case W = X. For a map W → X we often use the convention Since our methods are quite general, we expect that the results contained in this thesis apply in other contexts such as theétale topology on schemes, or for twisting different kinds of relative dualities in mathematics. By a gerbe, we shall always mean a O × -gerbe with trivial band on a complex analytic space. On a complex analytic space X, we use Mod(X) to denote the stack of abelian categories of O X -modules, which has global sections Mod(X). We use Coh(X) (QCoh(X)) to denote the stack of abelian categories of coherent (quasi-coherent) O X -modules, which has global sections the abelian category Coh(X) (QCoh(X)). We will also need to consider sheaves on gerbes X → X. For every analytic space Z mapping to X by f : Z → X, we consider the stack of functors f −1 X → f −1 Mod(X) and their natural transformations. This gives the stack of abelian categories of O X -modules Mod(X) → X, and similarly we have the stack of abelian categories of coherent (quasi-coherent) O X -modules Coh(X) → X (QCoh(X) → X) and we call their global sections the abelian categories Mod(X) and Coh(X) (QCoh(X)) respectively. We use D * (X) and D * (X) to denote the derived categories of Mod(X) and Mod(X). When nothing appears in the location of the symbol * , or * = ∅, this refers to the unbounded derived categories. When * = −, this refers to the bounded above derived category, and when * = b, this refers to the bounded derived category. Also D * c (X) and D * c (X) or D * qc (X) and D * qc (X) refer to the derived categories of coherent and quasi-coherent sheaves. For a discussion of quasi-coherence in the complex analytic context, see [4]. The sheaves of weight k on a gerbe will be denoted by Mod(X, k), Coh(X, k), QCoh(X, k) with global sections Mod(X, k), Coh(X, k), QCoh(X, k), with their associated derived categories D * (X, k), D * c (X, k), and D * qc (X, k).

Mukai's Insight
In this section we review some well known facts, mainly due to Mukai [29]. None of the material in this section is original. Given two complex analytic spaces M and N over B and an element In other words, we have an isomorphism This isomorphism uses the symmetry property of the tensor product. Given two complex analytic spaces M and N and an element K ∈ D(M × B N) we denote by Φ . Now, as noted in [2], these transforms enjoy the following locality property. Fix flat maps h : S → B and h ′ : S ′ → S. For us the most important case will be the inclusions of an open sets. Let M S , N S , M S ′ , N S ′ denote the fiber products. Let K S , and K S ′ denote the derived pullbacks of K to The integral transform has the following convolution property (see [29] or [30, Proposition 11.1]): If M, N and P are complex analytic spaces over B and K ∈ D b (M × B N) and L ∈ D b (N × B P ), then one has a natural isomorphism of functors This isomorphism of functors is compatible with base change, in the sense that if L S and K S are the derived pullbacks of L and K inside D(M S × S N S ) and D(N S × S P S ), with respect to the maps h M N : [23], we have that • for any f : N → P over B we have the natural isomorphism of functors For a gerbe X, it is a generally known fact that the category Mod(X) has enough injective and enough flat objects, see for example [14] and [26]. Indeed, following [26], if a : U → X is an atlas, and M is a sheaf on X, then we can find an injection a * M → I where I is an injective O U -module. We can also find a surjection where F is a flat O U -module. Therefore, we have a sequence of injections where a * I is injective and a sequence of surjections where a ! F is flat.

Some Functors
Recall that for X and Y complex analytic spaces, a gerbe Y → Y and a morphism f : We will often use the left exact functor f * , and the right exact functor f * The first of these is defined by the composition The second is defined by the composition Also if we have two gerbes X → X and Y → X over the same space we can define the gerbe X ⊗ Y → X to be the O × X gerbe induced from the If we are given M ∈ ob(Mod(X)) and N ∈ ob(Mod(Y)), then we can form This leads, for N ∈ ob(Mod(X)) to the right exact functor T N This is given by the composition

Some Derived Functors
If f is flat, we have the derived functor obtained by applying f * term by term to complexes. This functor obviously preserves the three types of boundedness. For a general map f we have the derived functors and

Presentations of Gerbes
Let W be an analytic space. A presentation of a gerbe on W is defined to be the following.
• an analytic space U mapping to W by a surjective local isomorphism (atlas) to W : a : U → W, • a line bundle L over U× W U; we will denote the left and right O−module structures on L by • that there is an equality of isomorphisms We will denote the presentation of a gerbe over W defined in this way by Here, we suppress the information of the map a : U → W in the notation, assuming that this map is clear from the context. and For completeness we note that a weak equivalence of two presentations of gerbes (L,θ,η) W and (L ′ ,θ ′ ,η ′ ) W over a : U → W is a pair (Q, τ ) consisting of Satisfying the equality of isomorphisms and the equality of isomorphisms where we have inserted the natural isomorphisms

Sheaves on Presentations of Gerbes
A sheaf of weight k on a presentation of a gerbe with atlas U → W is • which satisfies the extra condition that the following diagram commutes The sheaves on a presentation of a gerbe form an abelian category in the obvious way, a morphism between such sheaves S and T is simply a map S → T of sheaves on U such that the diagram below commutes (3.11) In this way we get abelian categories Mod( (L,θ,η) W, k) and Coh( (L,θ,η) W, k). They can also be seen as the categories of cartesian sheaves on the appropriate semi-simplicial space. For any presentation (L,θ,η) W of a gerbe W → W , we have isomorphisms and Coh( (L,θ,η) W, k) → Coh(W, k).

Derived Pushforward, Pullback, and Tensor Product
As mentioned before, there are some abstract ways to see that the category of sheaves on a gerbe possess injective and flat objects. To improve this slightly, we would like to see that there are flat and injective objects in Mod( (L,θ,η) W ) ∼ = Mod(W) which map to the same type of objects in Mod(U). For this purpose consider the commutative diagrams Notice that for any sheaf M of weight k on W equation 3.8 gives the existence of an injection a * M → a * a * I for an injective sheaf I on U. Flat base change then implies the existence of an injection in Mod( (L,θ,η) W ) Suppose now that we have a commutative diagram where A is an atlas for M, B is an atlas for N, and that the map h induces a maph : M → N. Furthermore, suppose we are given a gerbe N on N. Chose a gerbe presentation (L,θ,η) N of N using B and also consider the pullback presentation (h * L,h * θ,h * η) N of h * N on the atlas A. Then using resolutions as we have described above (simultaneously flat or injective on both the atlas and on the presentation), we see that the following diagrams commute Also for any object S of D(N, c) and its associated object a * S of D( (L,θ,η) N, c) we have the following commutative diagram Notice, that when thinking in terms of this presentation, we have for every k ∈ Z an adjoint pair of functors (a * k , a k * ), Here a k * (S) = p 0 * (L k ⊗ p * 1 S). We will later use this in the form when we will be interested in the case k = 1. Also note that a * k is the obvious forgetful map.

Constructing New Presentations From Old
and U the pullback of this open cover by π. In other words we will use the atlas We will consider the space X to be presented by the diagram where the two maps are the obvious projections or in other words Here we use the maps p n i 0 ,...,in : U n+1 → U n given by p n i 0 ,...,i n−1 (u 0 , . . . , u n ) = (u i 0 , . . . , u i n−1 ) We will not need the degeneracy maps too much, but they are the obvious maps induced by the diagonal. We will consider the sheaf of groups Aut(X/B) on B. A section of this sheaf over an open set U ⊂ B is just an automorphism of π −1 (U) which commutes with the projection to the base.
To every element f ofŽ 1 B (B, Aut(X/B)) we will associate a new analytic space π f : X f → B.
Let π n : U n → B be the obvious projection maps and consider the au- Therefore we have the equality p a cd • f a cd = f • p a cd , and of course f = f 01 . The fact that f is inŽ 1 (B, B, Aut(X/B)) just means that f 2 01 • f 2 12 = f 2 02 and that f • ∆ = ∆. We will often just write this more conveniently as The map f defines a new analytic space X f and the map π f : X f → B is now defined by the presentation (where we take the standard inclusions for the top and bottom arrows) via the isomorphisms . This is perhaps more transparent in the other notation where we have the commutative diagram In what follows, we will usually use the top row as a presentation of X f . Of course, an equivalence of two presentations of the twisted fibrations X f and X f ′ , given by f and f ′ , is given by an automorphism h of U which satisfies π • h = π, and if we let h c denote the restriction of h to U × X U, via p c : U × X U → U, This will often be written as This defines a new simplicial manifold (U • ) f whose first terms look like U and the degeneracy maps are induced by f and the diagonal. We have an isomorphism of simplicial manifolds

Presentations of Gerbes on Twisted Fibrations
In order to define presentations of gerbes on the space X f we will need to consider the terms coming from the resolution of X f over B described above. Due to the above remark, we see that a presentation of an O × gerbe on X f is defined by a line bundle L on U × X U together with isomorphisms which satisfy the following conditions Here we use to denote the obvious maps.
In other words, a presentation of a gerbe on a twisted version of X is a quadruple (f : satisfying the above compatibilities. We often write these as the isomorphisms Finally, let us comment on the notion of weak equivalence of two presentations (L,θ,η) X f and (L ′ ,θ ′ ,η ′ ) X f ′ of gerbes on twisted versions of X over the same cover. Note that any equivalence h of two twisted fibrations X f and X f ′ gives rise to a map Ξ(h) between presentations of gerbes on X f ′ and presentations of gerbes on X f . Guided by this observation we define a weak equivalence between (L,θ,η) X f and (L ′ ,θ ′ ,η ′ ) X f ′ to be a triple (h; Q, τ ), where h is an equivalence between f and f ′ , and the pair (Q, τ ) is an equivalence between the two gerbe presentations (L, θ, η) and Ξ(h)(L ′ , θ ′ , η ′ ) on X f . The map Ξ(h) is defined on objects by

Classification of Gerbes and the Leray-Serre Spectral Sequence
Isomorphism classes of gerbes on the space X f are classified by (the limit over covers C of X f ) of the second total cohomology group of the double complexČ where p ≥ 0 and q ≥ 0. We point out that the open sets in C are small and have nothing to do with U and U f . We will write this complex with p increasing in the horizontal direction and q increasing in the vertical direction. The rows only have cohomology in degree zero, so this double complex is equivalent to the kernel column of the first horizontal map: the column In other words, we have The right hand should be thought of as follows: we can think of a gerbe on X f as a gerbe on U f = (U f ) 0 together with a twisted line bundle for the difference of the pullback gerbes on (U f ) 1 (an isomorphism of the two pullback gerbes) and a relation between the pullback of these twisted line bundles to (U f ) 2 which itself goes to the identity in (U f ) 3 . In fact the double . In other words, we think of the Leray-Serre spectral sequence for π f : X f → B as a spectral sequence associated with a "Čech toČech" double complex where one of the covers is by big opens and one is by small opens.
So, far everything we have said is general and applies to any fibration c : W → B, where we would replace U • f by the fiber products over W of the disjoint union of the pullback by c of a cover of the base. Now the isomorphism of simplicial manifolds (U f ) • ∼ = (U • ) f says that we can classify gerbes on X f through data defined on X; we have Of course we will also use the isomorphism of simplicial manifolds

Reinterpretation and Duality
A fibration X → B defines a stack of abelian categories on B, by pushing forward the stack Mod(X) from X. To an open set U in B we associate the stack of abelian categories Mod(π −1 (U)). On the other hand, given any gerbe over a twisted version of X, which trivializes along the pullback of an open cover of B, we can similarly associate a stack of abelian categories on B. Below, we explicitly re-express the data defining a presentation of a gerbe over a twisted version of X in terms of the descent data of a stack of abelian categories on B, which is locally on the base given as the stack on B which was associated to X. Of course, this new stack defined by this descent data is isomorphic to the stack of sheaves on the gerbe corresponding to the presentation. Consider the sheaf of groupoids given by P ic(X/B) × Aut(X/B). The multiplication rule for this sheaf is In these terms, a gerbe presentation on X f consists of isomorphisms for which the diagram be the functor given on objects by We use θ ijk to denote the natural transformation Similarly, η i will denote the natural transformation Thus we have reinterpreted a presentation of a gerbe as a collection functors and natural transformations satisfying some commutative diagrams.
Lemma 5.1 A presentation of a gerbe over a twisted version of X → B, on the atlas U → X, defines a collection of functors A ij and natural transformations θ ijk and η i as above, so that the following diagrams commute.
The commutativity of these diagrams follows from equations 4.3,4.4 and 4.5. Indeed we need simply to observe that Notice that if (A, θ, η) are as above and A ′ ∈ Aut(Coh(U 1 )) is a functor that is isomorphic to A in a way compatible with restriction on base, then there exist canonical natural transformations θ ′ , and η ′ such that (A ′ , θ ′ , η ′ ) satisfy the same diagrams as do (A, θ, η). Therefore, we could have taken A to be any automorphism of stacks isomorphic to T L • f * .
is an invertible natural transformation which satisfies the equality of natural transformations as natural transformations Indeed, we can recognize the equation 5.4 as containing the maps Remark 5.4 The lemma 5.1 gives a fully faithful functor from the category whose objects are presentations (L,θ,η) X f of gerbes on twisted versions of X → B using the atlas U → X and whose morphisms are weak/strong equivalences, to the category whose objects are triples (A, θ, η) and whose morphisms are weak/strong equivalences.
Remark 5.5 Any triple (A, θ, η) defines an abelian category whose objects are sheaves S on U, along with isomorphisms

commutes, and morphisms are maps S → T such that the diagram
Any strong equivalence between triples (A, θ, η) and (A ′ , θ ′ , η ′ ) induces an isomorphism of the associated abelian categories. If the triple (A, θ, η) comes from a gerbe presentation (L,θ,η) X f then the category we have described is isomorphic to the category of sheaves of weight (−1) on the presentation (L,θ,η) X f . In the future, we will make this identification implicit.
Definition 5.6 Suppose we are given maps π : X → B and ρ : Y → B, and We call a gerbe presentation (L,θ,η) X f Φ-compatible if we have where A ∈ Aut(Coh(U 1 )) corresponds to (M,θ,η) X f and C ∈ Aut(Coh(V 1 )), is isomorphic to the functor S → M ⊗ g * S, for some line bundle M, and automorphism g preserving the projection to the base.
We call a gerbe X → B over a twisted version of X Φ-compatible if it admits a Φ-compatible presentation (L,θ,η) X f . given by Υ :

The Dual Gerbe on the Dual Fibration
We will denote this on elements A ij by the equation The functor Υ(A ij ) satisfies for some line bundles M ij on ρ −1 (U ij ), and some fiber preserving automorphisms g ij of ρ −1 (U ij ). Here the isomorphism respects the pullback along open sets in the base. In other words, this is an isomorphism of automorphisms of the pushforward of the stack of coherent sheaves on Y to the base B.
Let Aut(Mod(U 1 )) Φ be the sub group-groupoid of Aut(Mod(U 1 )) consisting of the elements A for which Υ(A) is isomorphic to T M • g * for some line bundle M and projection preserving automorphism g. We get an induced isomorphism of categories, where the morphisms are strong equivalences By applying Υ to the diagram 5.1 we obtain the following commutative diagram.
This is not quite what we want, but if we attach to this diagram three other commuting squares, we get the outer diagram below, which is the one that we want.
We now rewrite this, defining T ikl to be the right hand side of the above diagram: Similarly by applying Υ to the diagrams 5.2 and 5.3 we get the diagrams By applying Υ to the natural transformation η i : 1 ii ⇒ A ii , we can define ζ i as the composition 1 ii ⇒ Υ(1 ii ) ⇒ Υ(A ii ). Using ζ i : 1 ii ⇒ Υ(A ii ) we can complete the above diagram as In other words, we obtain the commutative diagrams (5.14) Notice that we end up with the same type of diagrams 5.9, 5.14, 5.15 that we started with on the other side 5.1, 5.2, 5.3 . Finally, we apply the diagrams 5.9, 5.14, 5.15 to the structure sheaves of points C y on Y , and to the structure sheaf O Y of Y . The application to points shows that g is a cocycle. Application to the structure sheaf gives a presentation of a gerbe on Y g . This gives rise to a map an automorphism g : V 1 → V 1 , a line bundle M → V 1 , and isomorphisms Thus, we have found a presentation (g : V 1 → V 1 , M → V 1 , T, ζ) of a gerbe on a twisted version of Y . This gerbe presentation (M,T,ζ) Y g → Y g over a twisted version of Y , was found starting from the presentation of the gerbe (L,θ,η) X f → X f over a twisted version of X. We state this as the following Proof Above. ✷ Definition 5.8 Suppose X is a Φ-compatible gerbe on a twisted version of X and Y is the corresponding gerbe on a twisted version of Y produced as described above. Then we say that Y is Φ-dual to X. Y is defined by a Φ-dual presentation (M,T,ζ) Y g to a Φ-compatible presentation (L,θ,η) X f of X.
Notice that the definition of T is encoded in the diagram

The New Functor
It turns out that equations 5.6 and 5.19 secretly encode the description of a new functor. We have the following lemma.
Lemma 5.9 Suppose we are given compact, connected, complex manifolds X and Y , maps π : X → B and ρ : Y → B, where ρ is flat and an object flat over Y giving a derived equivalence Φ : D b c (Y ) → D b c (X) and a quasiinverse equivalence Ψ. Then for every Φ-compatible gerbe presentation (L,θ,η) X f and Φ-dual gerbe presentation (M,T,ζ) Y g , we can define a functor In order to see this, we must first manipulate the equation 5.19 a little. By applying Φ on the left to the diagram 5.19 and attaching a commutative lower square, we get Taking the outer part of this we arrive at the commutative diagram Remark 5.10 Every corner of this square is a functor implemented by a sheaf on the appropriate product. If we could find unique sheaf isomorphisms implementing the natural equivalences indicated by the arrows, this diagram would describe a sheaf of weight (−1, 1) on the product of our gerbe presentations and it would be easy to show that such a sheaf implements an equivalence using the convolution product. However, it is not clear to the author how to produce these isomorphisms, and so we must follow a different path.
We will sometimes write this functor as Here we define φ(S) = H 0 (Φ U (S)) and φ(κ) : . For the sake of sanity, we will change notation, skip the second and second to last term. We write the object of Mod( (M,T,ζ) Y g , −1) as where λ ij is defined by the composition The fact that this is an object of Mod( (M,T,ζ) Y g , −1) rests on the commutativity of the outer square in the following: We can see that the outer square is commutative by checking that all the inner squares commute. The lower right square is commutative as a the result of applying Φ followed by H 0 to the descent diagram for ({S i }, {κ ij }). This descent diagram can be found as diagram 5.5 where A must be replaced by Υ(A). The square to the left of the lower right square is commutative using 5.21. The other squares of the diagram are clearly commutative. Therefore the functor φ is well defined. We define Φ to be the right derived functor R φ of the left exact functor φ.
• D b c (S, k) admits a Serre functor defined by • The structure sheaves of points on S viewed as sheaves of weight k, define a spanning class for D b c (S, k). We now recall a general criterion, due to Bondal-Orlov, and Bridgeland, for a functor F : A → B to be an equivalence of linear triangulated categories.
Theorem [7,8,9] Assume that A and B have Serre functors S A , S B , that A = 0, A has a spanning class C, B is indecomposable, and that F : A → B has left and right adjoints. Then F is an equivalence if and it intertwines the Serre functors: on all elements x ∈ C in the spanning class.
Theorem 5.11 Let X and Y be compact, connected, complex manifolds and π : X → B and ρ : Y → B be maps of complex analytic spaces, with ρ flat and P ∈ Coh . Let X be a Φ-compatible gerbe on a twisted version of X. Let Y be a Φ-dual gerbe to X (so Y lives over a twisted version of Y ). Then we have an equivalence of categories Proof. First of all, note that our functor Φ is a right-derived functor, and hence has a left adjoint. Since D b c (Y, −1) and D b c (X, −1), both have Serre functors, Φ therefore has a right adjoint as well. It is obvious that D b c (Y, −1) is not zero, and that D b c (X, −1) is indecomposable, and it has already been noted [19] that the points give a spanning class of D b c (Y, −1). Therefore, we simply need to show that Φ is orthogonal on this spanning class, and that it intertwines the Serre functors on this spanning class.
We claim that Φ is orthogonal on the spanning class consisting of points in Y g . Let v 1 and v 2 be any two points in V. Then we know that is an isomorphism, where we have used the isomorphism 5.23. Now the commutative diagram induces the following commutative diagram where the vertical arrows are isomorphisms, with inverses given by the various pushforward functors, see 5.23 and 5.24.
Since Φ is an equivalence, the bottom is an isomorphism and so we are done. Also using the fact that Φ is an equivalence, we know that Φ intertwines the Serre functors of D b c (Y ) and D b c (X). In other words Now pulling back by the cover U → B and using the property 3.2 we see that for all sheaves S on V we have Now we claim that Φ intertwines the Serre functors of D b c (Y, −1) and D b c (X, −1) on the spanning class. Consider a point v of V. Observe there is a sheaf S = C v , with the properties Rb * S ∼ = C b(v) , Rb g * S ∼ = C bg(v) , and Rb * ,−1 S ∼ = C bg (v),−1 . Then if we let T = Φ U (S), we have Ra * T ∼ = Φ(C b(v) ), and Ra * ,−1 T ∼ = Φ(C bg(v),−1 ). This follows from equations 5.23 and 5.24.
Moving forward, we have, and ) and by 5.24 and the fact that Φ intertwines the classical Serre functors (5.25) however and finally −1 )) Therefore, the Serre functors are intertwined on the spanning class by Φ, so we have checked all the criteria and hence Φ is an equivalence. ✷

Alternative Method
In this subsection, we give another method for proving the equivalence. This method applies in different situations, allowing us to drop various coherence, smoothness, and compactness assumptions. We were convinced of the possibility/usefulness of such a method due to comments of D. Arinkin who kindly provided his insights to us after a reading of the thesis. In the above, we have given an assignment A which takes a pair of shifted sheaves P, Q ∈ D c (X × B Y ) giving inverse equivalences, and a compatible gerbe presentation X = (L,θ,η) X f to an equivalence D( Y ) → D( X), where Y is the dual presentation to X defined by the pair (P, Q). We encode this assignment as Observe that this assignment has the following properties: • Given isomorphisms P ∼ = P ′ , and Q ∼ = Q ′ , the following diagram is commutative D( Y ( X; P, Q), −1) • Suppose we are given shifted coherent sheaves P 1 , Q 1 on X × B Y , and P 2 , Q 2 on Y × B Z giving inverse equivalences, and a (P 1 , Q 1 ) compatible presentation X. Let Y denote Y ( X; P 1 , Q 1 ), and suppose that Y is (P 2 , Q 2 ) compatible. Let Z denote Z( Y ; P 2 , Q 2 ). Then if R = P 1 * P 2 and S = Q 2 * Q 1 are shifted sheaves, and we let the inverse equivalences given by R and S fit into the following commutative diagram: Proof. By the above properties of A, the functors associated to P and Q compose to a functors equivalent to the identity, since P and Q convolve to sheaves isomorphic to the diagonals. ✷ Remark 5.1 Using the definition of quasi-coherence found in [4], we expect that one would also get equivalences for * = b, −.

An Application
In this chapter we provide an application. We would like to emphasize that the geometric limitations we impose in the following are only necessary in order to allow for the setup of Donagi and Pantev to be reproduced via our setup. The situations that we will consider in the following are not the most general applications that one could imagine of our main theorem, even in the context of elliptic fibrations.
In the first section, we look at an analogue of Donagi and Pantev's setup for complex torus fibrations, and prove a general corollary of our main theorem. In the second section, we apply this corollary to prove their conjectures.

Complex Torus Fibrations
Consider a complex torus fibration X → B with section σ : B → X. By this we just mean that away from a closed co-dimension one analytic subset of B, the fiber of π is a complex torus. Away from the singular fibers this is an analytic group space over B, using the section as an identity. Therefore over the complement of the singular fibers, the translation by a section (relative to σ) defines an automorphism of the map X → B. Definition 6.1 Let us call a complex torus fibration with section X → B reasonable if X is a compact, connected complex manifold, the map to B is flat and if the translation by any local section U → X extends uniquely to an automorphism of the map X × B U → U.
Let π : X → B and ρ : Y → B be reasonable complex torus fibrations with sections over a common base B. We use the section to give a group structure to the sheaves of sections. A sheaf S on X × B Y is called a bi-extension if for any local sections σ : U → X and ψ : U → Y we have that S| σ×ρ −1 (U ) and S| π −1 (U )×ψ are line bundles and we have isomorphisms on X × U Y × U ψ and these isomorphisms satisfy certain natural compatibilities. In particular notice this implies that for small open sets U ⊂ B and sections σ of X over U, and ψ of Y over U that if f (and g) represent the translation action of σ on X (ψ on Y ) we have Now if U has no non-trivial line bundles, then the line bundle on U given by ((σ, ψ) * S| π −1 (U )×ρ −1 (U ) ) can be trivialized, so we get an isomorphism Similarly there exists an isomorphism, Furthermore the bi-extension structure gives us isomorphisms Suppose that π : X → B and ρ : Y → B are reasonable complex torus fibrations with section over the same base B and let us denote by X and Y the sheaves of sections of X and Y respectively. By the above injections of sheaves, the sheaves X and Y become sub-sheaves of groups in Aut X/B and Aut Y /B respectively. We assume that there is a coherent Poincaré sheaf P living on X × B Y , flat over Y that implements an equivalence of derived categories Φ : D b c (Y ) → D b c (X), and is a bi-extension. Consider two elements α ∈ H 1 (B, Y) and β ∈ H 1 (B, X ). Let f ∈Ž 1 (B, Aut X/B ) and g ∈Ž 1 (B, Aut Y /B ) be the automorphisms given by translation by some representatives for β and α respectively. We now define a map Indeed we can send a local section γ i over U i to the equivalence class Let g ij be the automorphism of ρ −1 (U ij ) corresponding to the section α ij . Now, because of equation 6.2 (following from the reasonable nature of the torus fibrations and the bi-extension property of P) we have, for sections γ i and γ j agreeing on the overlap in . This shows the map that we have described is well defined. Similarly, we have a map By taking the induced map on cohomology, we denote by S α (β) the image of β under the map Similarly, we denote by S β (α) the image of α under the map Let us further assume that the images of S α (β) and S β (α) (via the differentials d g;2 and d f ;2 described in section ) vanish in H 3 (B, O × ) and that the image of H 2 (B, O × ) vanishes in both H 2 (X β , O × ) and H 2 (X α , O × ). Then, according to the spectral sequence for the fibrations we find unique gerbes α X β → X β and β Y α → Y α which correspond to S β (α) and S α (β) respectively. We then have the following corollary of the main theorem. For certain elliptic fibrations X = Y with some possible further restrictions on the pair (β, α), this fact has already been proven by Donagi and Pantev [19]. For some other cases of β and α this was conjectured. The following corollary proves the conjectures (see e.g. conjecture 2.19) from [19] and as well it gives an alternative proof of the main results in the paper [19].
Corollary 6.2 In the above circumstance, we have Proof.
In order to prove this we simply need to find gerbe presentations (L,θ,η) X f of −α X β and (M,T,ζ) Y g of β Y α which are dual in the sense of the main theorem. For then we will have We do this simply by picking a reasonable choice for (L,θ,η) X f and showing that its dual presentation represents β Y α . We know that the gerbe −α X β comes from the E ∞ term of the Leray-Serre spectral sequence, given by In where we have used equation 6.1. Consider the natural filtration Here ) are the classes representable by an element whose ) are the classes representable by an element whose components inČ 2 ( ∞ of the obvious modification L ′ ∈Č 1 ((U 1 ) f , O × ) of L given by choosing an element inČ 0 ((U 1 ) f , O × ) which trivializes θ. Notice, that we have just replaced L by an isomorphic line bundle L ′ , in other words, In the other direction consider the following set theoretic splitting of the surjection Given an element [[L]] ∈ E 1,1 ∞ , we know that [L] goes to zero in H 1 ((U 2 ) f , O × ) and therefore the image of L inČ 1 (( ∞ , θ goes to the trivial element inČ 0 ((U 3 ) f , O × ). Therefore, the pair (L, θ) gives an element in is trivial (the pullback of a gerbes from the base is trivial), the inverse map (the set theoretic splitting) is actually a group homomorphism and we have written explicitly the two maps in the isomorphism We now apply the above in the case that L = P| (U 1 ) f ×−α , in other words L ij = P| (U 1 ) f ×−α ij . We have canonical trivializations η i : O → L ii , and therefore we have produced a presentation of a gerbe (L,θ,η) X f on X Now the gerbe presentation (L,θ,η) X f corresponds, according to the proof of Lemma 5.1, to the element Pre-composing this by the Fourier-Mukai transform Now using the equations 3.4 and 6.3 we have Now we have, using equations 3.5 and 6.4 we have By putting all this together, we arrive at Equation 6.5 shows that the presentation (L,θ,η) X f is Φ P -compatible (see definition 5.6), and therefore the gerbe −α X β is Φ P -compatible and so we can calculate the dual gerbe on a twisted version of Y . Therefore, by the lemma we have a dual gerbe, on the presentation Y g of Y α , given by M = g * (P| β×V 1 ) → V 1 , along with some isomorphisms T ijk : M jk ⊗ g * jk M ij → M ik , and ζ i : O → M ii satisfying the diagrams of a gerbe presentation. According to the above discussion, the equivalence class of this gerbe on Y α comes from the term in E 1,1 ∞ given by [ . This is the gerbe on Y α which comes from β ∈ H 1 (B, X ) and so we are done. ✷ Remark 6.1 Clearly, we can also prove a more general statement by invoking Theorem 5.12, we leave the recording of this statement to the interested reader.

The Conjecture of Donagi and Pantev
In this section we use a special case of the corollary 6.2 and use it to reprove the main results in [19] as well as the following conjecture [19] of Donagi and Pantev, which was proven in [19] in many important special cases. Conjecture 6.3 Let X be a complex manifold elliptically fibered with at worst I 1 fibers over a normal analytic variety B such that Let α, β ∈ X an (X) be complementary elements (see [19] for the definition). Then there exists an equivalence of the bounded derived categories of sheaves of weights ±1 on α X β and β X α respectively.

Proof
Let Y = X and let X be the sheaf of sections of X, considered as a sheaf of groups via a choice of a section σ. We have two flat morphisms, π : X → B, and ρ : X → B giving X two different structures of a reasonable torus fibration over B. We then get an isomorphism Consider the rank one divisorial sheaf on X × B X defined as where ̟ : X × B X → B is the natural projection. It gives an equivalence of derived categories Φ : D b c (X) → D b c (X) (see for example the paper of Bridgeland and Maciocia [10]). Furthermore, as explained in [19], P is a bi-extension. The fact that α and β are complimentary (the triviality of the pairing < S α (β), S β (α) > from [19]) is equivalent to the vanishing of the images of S α (β) and S β (α) in H 3 (B, O × ) as explained in [19]. Now an application of Corollary 6.2 immediately produces an isomorphism as in 6.6 and hence proves the conjecture 6.3 of Donagi and Pantev [19]. ✷

Conclusions and Speculations
In this section, we comment on some possible future generalizations of our main theorem. After this, we indicate a few theorems that we expect as consequences or analogues of the results of this thesis. First of all, we expect to be able to dispense with the compactness and smoothness assumptions of our varieties, as well as to be able to extend our main result to the case where the fibrations are not flat, and perhaps the Poincaré sheaf is a more general object in the derived category of the fiber product. Certainly, we could replace our Serre functor with a suitable object in a singular situation. Compactness assumptions can be dispensed with by considering derived categories of compactly supported sheaves. The key ingredient needed for generalizations seems to be to find a diagram of isomorphisms of sheaves or complexes of sheaves which induce the diagram of natural equivalences 5. 21. In many cases this should be possible using the constraints on the Poincaré object coming from the fact that it implements an isomorphism. Another strategy towards finding an object on the fiber product of the two gerbes is to use the technique of cohomological descent for gluing objects in the differential graded categories: locally the object is just the Poincaré object, and the diagram of natural equivalences 5.21 becomes a descent data diagram in the differential graded category of sheaves on the fiber product.
Ideally, there would also be a presentation free description of the twisting of derived equivalences. A scheme to do precisely this was suggested to us by D. Arinkin during a discussion of the original version of the thesis. It is an extension of his duality for group stacks perspective, as found in the appendix to [19]. We hope that future results generalizing those of our main theorem will find presentation free descriptions.
In the search for applications of our main theorem to fibrations where the generic fiber has no group structure, a natural place to start is K3 fibrations. However, as we now explain, the most naive attempts seem to lead back to scenarios which are already understood. First, we start with two K3 surfaces W and Z such that Z is a fine moduli space for stable sheaves of Mukai vector (r, H, s) on W . If L is any line bundle on W , then the map acts on the Mukai vector by T L (r, H, s) = (r, rc 1 (L) + H, s + (c 1 (L), H) + 1 2 r(c 1 (L) 2 ). Since c 1 (L) is non-torsion, r must then be zero in order that T L preserves the moduli problem, and can be used to build gerbes with the derived equivalence between Z and W . Unfortunately, this together with the fact that (r, H, s) 2 = 0 implies that H 2 = 0. Now we can find a non-trivial line bundle M such that c 1 (M) = H. Then M defines an elliptic fibration W → P 1 . The sheaves on W corresponding to points on Z will all be supported on the fibers of this elliptic fibration. This argument seems to go through in the relative case (where (r, H, s) now becomes a relative Mukai vector) and this shows that a K3 fibration W → B admitting gerbes compatible with a derived equivalence (which are not pull-backs from B) becomes an elliptic fibration over a P 1 fibration over B, and the derived equivalence respects this elliptic fibration structure. If it works, this reasoning will only be bi-rational, and refining it to an actual geometric statement may lead to new applications to K3 fibrations. Also, note that if we replace our K3 fibration with a, say, abelian surface fibration, then we could have r be nonzero, provided that L was r-torsion, and hence we expect a straightforward application of our main theorem in that case. A perhaps more obvious application concerns the moduli spaces of vector bundles on curves. For concreteness, let C be a genus 2 hyperelliptic curve. Then from the work of Desale and Ramanan [17] we recall that the moduli space of (isomorphism classes of stable) vector bundles of rank 2 and fixed odd determinant of odd degree is isomorphic to the intersection X = Q 1 ∩ Q 2 of two quadric four-folds in P 5 . Bondal and Orlov, in [7] give an explanation in terms of derived categories. They produce a vector bundle V on C × X that implements a full and faithful embedding D b c (C) → D b c (X). Finally they Embed E 2 into projective space, this gives us a line bundle O(1) on E 2 . Fix a class {n ij } ∈Ž 1 (B 1 , Z) for some small open cover {U i } of B 1 . Consider an O × −gerbe X on X presented by the collection of line bundles {µ * O(n ij ) → π −1 (U ij )}. Now conjugating the tensorization by µ * O(n ij ) by the (appropriately restricted) transformations Φ U ij yields automorphisms g n ij of ρ −1 (U ij ) = U ij × F ∨ 1 × S 2 . These are symplectomorphisms of ρ −1 (U ij ) which are constant on the U ij × F ∨ 1 factor, and given by n ij compositions of a Dehn twist g on the factor S 2 . Using them, we can produce a new symplectic 4-manifold Y .
Then, by analogy with the main theorem, we can conjecture that Notice that the automorphisms destroy the second S 1 fibration structure, but does nothing to the first one. By contrast, if the O(1) was replaced by a line bundle flat along the F 2 direction, we would expect to keep both S 1 fibration structures on the symplectic side, fixing the first one and turning the second into a non-trivial principal S 1 bundle.
Finally, we would like to mention a recent paper of Donagi and Pantev on the duality between the Hitchin integrable systems associated to a simple complex Lie group and its Langlands dual. There, they prove a derived equivalence between the bounded, compactly supported, coherent derived categories of the commutative group stacks L Higgs and Higgs, over a space B − ∆. Here ∆ is a discriminant locus over which the Hitchin fibration has singular fibers. They remark there that the results in the thesis can be used in their proof, and further should be useful to extend the duality over all of B. For a more complete explanation, we refer to their paper [20].