Duality and Equivalence of Module Categories in Noncommutative Geometry

Centre de Recherches Math´ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Jonathan Block

In memory of Raoul

Abstract. We develop a general framework to describe dualities from algebraic, diﬀerential, and noncommutative geometry, as well as physics. We pursue a relationship between the Baum – Connes conjecture in operator K- theory and derived equivalence statements in algebraic geometry and physics. We associate to certain data, reminiscent of spectral triple data, a diﬀerential graded category in such a way that we can recover the derived category of coherent sheaves on a complex manifold.

Introduction In various geometric contexts, there are duality statements that are expressed in terms of appropriate categories of modules. We have in mind, for example, the Baum – Connes conjecture from noncommutative geometry, T-duality and Mirror symmetry from complex geometry and mathematical physics. This is the first in a series of papers that sets up a framework to study and unify these dualities from a noncommutative geometric point of view. We also view this project as an attempt to connect the noncommutative geometry of Connes, [6] with the categorical approach to noncommutative geometry, represented for example by Manin and Kontsevich. Traditionally, the complex structure is encoded in the sheaf of holomorphic functions. However, for situations we have in mind coming from noncommutative geometry, one can not use local types of constructions, and we are left only with global differential geometric ones. A convenient setting to talk about integrability of geometric structures and the integrability of geometric structures on their modules is that of a differential graded algebra and more generally, a curved differential graded algebra. Thus, for example, a complex structure on a manifold is encoded 0, in its Dolbeault algebra A =( •(X), ∂¯), and a holomorphic vector bundle can be viewed as the data of a finitelyA generated projective module over 0,0 together with A 2000 Mathematics Subject Classification. Primary 58B34; Secondary 18E30, 19K35, 46L87, 58J42. J.B. partially supported by NSF grant DMS02-04558. This is the final form of the paper.

c 2010 American Mathematical Society ! 1 2 J. BLOCK a flat ∂¯-connection. Similarly, holomorphic gerbes can be encoded in terms of a curved differential graded algebra with non-trivial curvature. Curved dgas appear naturally in the context of matrix factorizations and Laundau – Ginzburg models, [11,21]. Indeed, these fit very easily into our framework. Of course, one is interested in more modules than just the finitely generated projective ones. In algebraic geometry, the notion of coherent module is funda- mental. In contrast to projective algebraic geometry however, not every coherent sheaf has a resolution by vector bundles; they only locally have such resolutions. Toledo and Tong, [28,29], handled this issue by introducing twisted complexes. Our construction is a global differential geometric version of theirs. We have found the language of differential graded categories to be useful, [5, 10, 18]. In particular, for a curved dga A we construct a very natural differential graded category A which can then be derived. The desiderata of such a category are P it should be large enough to contain in a natural way the coherent holo- • morphic sheaves (in the case of the Dolbeault algebra), and it should be flexible enough to allow for some of Grothendieck’s six oper- • ations, so that we can prove Mukai type duality statements.

The reason for introducing A is that the ordinary category of dg-modules over the Dolbeault dga has the wrongP homological algebra; it has the wrong notion of quasi-isomorphism. A morphism between complexes of holomorphic vector bundles considered as dg-modules over the Dolbeault algebra is a quasi-isomorphism if it induces an isomorphism on the total complex formed by the gloabal sections of the Dolbeault algebra with values in the complexes of holomorphic vector bundles, which is isomorphic to their hypercohomology. On the other hand, and the PA modules over it have the correct notion of quasi-isomorphism. In particular, A is not an invariant of quasi-isomorphism of dga’s. To be sure, we would not wantP this. For example, the dga which is C in degree 0 and 0 otherwise is quasi-isomorphic to n n the Dolbeault algebra of CP . But CP has a much richer module category than anything C could provide. We show that the homotopy category of A where A is the Dolbeault algebra of a compact complex manifold X is equivalentP to the derived category of sheaves of X -modules with coherent cohomology. Our description of the coherent derived categoryO has recently been used by Bergman, [2] as models for B-model D-branes. To some extent, what we do is a synthesis of Kasparov’s KK-theory, [17] and of Toledo and Tong’s twisted complexes, [20, 28, 29].

In appreciation of Raoul Bott. I am always amazed by the profound impact that he had, and still has, on my life. During the time I was his student, I learned much more from him than mere mathematics. It was his huge personality, his magnanimous heart, his joy in life and his keen aesthetic that has had such a lasting eﬀect. I miss him.

Acknowledgements. We would like to thank Oren Ben-Bassat, Andre Cal- dararu, Calder Daenzer, Nigel Higson, Anton Kapustin, the referee, Steve Shnider, Betrand Toen and especially Tony Pantev for many conversations and much guid- ance regarding this project. DUALITY AND EQUIVALENCE 3

1. Baum – Connes and Fourier – Mukai There are two major motivations for our project. The ﬁrst is to have a general framework that will be useful in dealing with categories of modules that arise in geometry and physics. For example, we will apply our framework to construct categories of modules over symplectic manifolds. Second, as mentioned earlier in the introduction, this series of papers is meant to pursue a relationship between (1) the Baum – Connes conjecture in operator K-theory and (2) derived equivalence statements in algebraic geometry and physics. In particular, we plan to reﬁne, in certain cases, the Baum – Connes conjecture from a statement about isomorphism of two topological K-groups to a derived equivalence of categories consisting of modules with geometric structures, for example, coherent sheaves on complex manifolds. We will see that there are natural noncommutative geometric spaces that are derived equivalent to classical algebraic geometric objects. Let us explain the obvious formal analogies between (1) and (2). For simplicity let Γbe a discrete torsion-free group with compact BΓ. In this situation, the Baum – Connes conjecture says that an explicit map, called the assembly map,

(1.1) µ: K (BΓ) K (Cr∗Γ) ∗ → ∗ is an isomorphism. Here Cr∗Γdenotes the reduced group C∗-algebra of Γ. The assembly map can be described in the following way. On C(BΓ) C∗Γthere is a ⊗ r finitely generated projective right module P which can be defined as the sections of the bundle of Cr∗Γ-modules EΓ C∗Γ. ×Γ r This projective module is a “line bundle” over C(BΓ) Cr∗Γ. Here, C(X) denote the complex-valued continuous functions on a compact space⊗ X. The assembly map is the map defined by taking the Kasparov product with P over C(BΓ). This is some sort of index map.

µ: x KK(C(BΓ), C) x P KK(C,C∗Γ) ∈ %→ ∪ ∈ r where P KK(C,C(BΓ) Cr∗Γ) We now∈ describe Mukai⊗ duality in a way that makes it clear that it refines Baum – Connes. Now let X be a complex torus. Thus X = V/Λwhere V is a 2g g-dimensional complex vector space and Λ ∼= Z is a lattice in V . Let X∨ denote the dual complex torus. This can be described in a number of ways: as Pic0(X), the manifold of holomorphic line bundles on X with first • Chern class 0 (i.e., they are topologically trivial); as the moduli space of flat unitary line bundles on X. This is the same • as the space of irreducible unitary representations of π1(X), but it has a complex structure that depends on that of X; and most explicitly as V ∨/Λ∨ where Λ∨ is the dual lattice, • Λ∨ = v V ∨ Im v,λ Z λ Λ . { ∈ | ( )∈ ∀ ∈ } Here V ∨ consists of conjugate linear homomorphisms from V to C.

We note that X = BΛand that C(X∨) is canonically Cr∗Λ. Hence Baum – Connes predicts (and in fact it is classical in this case) that K (X) = K (Cr∗Λ) = ∗ ∼ ∗ ∼ K∗(X∨). 4 J. BLOCK

On X X∨ there is a canonical line bundle, P, the Poincar´ebundle, which is uniquely determined× by the following universal properties:

P X p = p where p X∨ and is therefore a line bundle on X. • | ×{ } ∼ ∈ P 0 X∨ is trivial. • |{ }× Now Mukai duality says that there is an equivalence of derived categories of coherent sheaves b b D (X) D (X∨) → induced by the functor

p2 (p1∗ P) F%→ ∗ F⊗ where pi are the two obvious projections. The induced map at the level of K0 is an isomorphism and is clearly a holomorphic version of the Baum – Connes Conjecture for the group Λ.

2. The dg-category of a curved dga PA 2.1. dg-categories. Definition 2.1. For complete definitions and facts regarding dg-categories, see [5, 10, 18, 19]. Fix a field k.Adifferential graded category (dg-category) is a category enriched over Z-graded complexes (over k) with differentials increasing degree. That is, a category is a dg-category if for x and y in Ob the hom set C C (x, y) C forms a Z-graded complex of k-vector spaces. Write ( •(x, y),d) for this complex, if we need to reference the degree or differential in theC complex. In addition, the composition, for x, y, z Ob ∈ C (y, z) (x, y) (x, z) C ⊗C →C is a morphism of complexes. Furthermore, there are obvious associativity and unit axioms. 2.2. Curved dgas. In many situations the integrability conditions are not expressed in terms of flatness but are defined in terms of other curvature conditions. This leads us to set up everything in the more general setting of curved dga’s. These are dga’s where d2 is not necessarily zero. Definition 2.2. A curved dga [23] (Schwarz [25] calls them Q-algebras) is a triple

A =( •, d, c) A where • is a (nonnegatively) graded algebra over a field k of characteristic 0, with a derivationA +1 d: • • A →A which satisfies the usual graded Leibniz relation but d2(a) = [c, a] where c 2 is a fixed element (the curvature). Furthermore we require the Bianchi ∈A identity dc = 0. Let us write for the degree 0 part of •, the “functions” of A. A A DUALITY AND EQUIVALENCE 5

A dga is the special case where c = 0. Note that c is part of the data and even if d2 = 0, that c might not be 0, and gives a non-dga example of a curved dga. The prototypical example of a curved dga is • M,End(E) , Ad ,F of differential forms on a manifold with values in the endomorphismsA of a vector∇ bundle E with connection and curvature F . ! ! " " ∇ 2.3. The dg-category . Our category consists of special types of A- PA PA modules. We start with a Z-graded right module E• over . A Definition 2.3. A Z-connection E is a k-linear map E: E• • E• • ⊗A A → ⊗A A of total degree one, which satisfies the usual Leibniz condition e E(eω)= E(e 1) ω +( 1) edω ⊗ − k Such a connection is determined! by its" value on E•. Let E be the component k k+1 k 0 1 2 of E such that E : E• E•− ; thus E = E + E + E + . It is → ⊗A A ··· clear that E1 is a connection on each component En in the ordinary sense (or the negative of a connection if n is odd) and that Ek is -linear for k = 1. A - Note that for a Z-connection E on E• over a curved dga A =( •, d, c), the A usual curvature E E is not -linear. Rather, we define the relative curvature to be the operator ◦ A F (e)=E E(e)+e c E ◦ · and this is -linear. A Definition 2.4. For a curved dga A =( •, d, c), we define the dg-category : A PA (1) An object E =(E•, E) in A, which we call a cohesive module, is a Z- P graded (but bounded in both directions) right module E• over which is finitely A generated and projective, together with a Z-connection E: E• • E• • ⊗A A → ⊗A A that satisfies the integrability condition that the relative curvature vanishes F (e)=E E(e)+e c =0 E ◦ · for all e E•. (2) The∈ morphisms of degree k, k(E ,E ) between two cohesive modules PA 1 2 E1 =(E1•, E1) and E2 =(E2•, E2) of degree k are

φ: E1• • E2• • of degree k and φ(ea)=φ(e)a a • { ⊗A A → ⊗A A | ∀ ∈A } with diﬀerential deﬁned in the standard way φ d(φ)(e)=E2 φ(e) ( 1)| |φ E1(e) − − Again, such a φ is determined by its! restriction" to E!1• and" if necessary we denote the component of φ that maps +k j j (2.1) E1• E2• − → ⊗A A by φj.

k k Thus A(E1,E2) = Hom (E1•,E2• •). P A ⊗A A Proposition 2.5. For A =( •, d, c) a curved dga, the category A is a dg- category. A P 6 J. BLOCK

This is clear from the following lemma.

Lemma 2.6. Let E1, E2 be cohesive modules over the curved dga A =( •, d, c). Then the differential defined above A +1 d: •(E ,E ) • (E ,E ) PA 1 2 →PA 1 2 satisfies d2 =0. 2.4. The homotopy category and triangulated structure. Given a dg- category , one can form the subcategory Z0 which has the same objects as and whoseC morphisms from an object x toC an object y are the degreeC 0 closed morphisms in (x, y). We also form∈ theC homotopy category∈C Ho which has the same objects as Cand whose morphisms are the 0th cohomology C C Ho (x, y)=H0 (x, y) . C C We define a shift functor on the category! A." For E =(E•, E) set E[1] = +1 P (E[1]•, E[1]) where E[1]• = E• and E[1] = E. It is easy to verify that E[1] . Next for E,F and φ Z0 (E,F−), define the cone of φ, Cone(φ)=∈ PA ∈PA ∈ PA (Cone(φ)•, Cφ) by F • Cone(φ)• =  ⊕  E[1]• and   F φ Cφ = 0 E[1] ' ( We then have a triangle of degree 0 closed morphisms φ (2.2) E F Cone(φ) E[1] −→ → → Proposition 2.7. Let A be a curved dga. Then the dg-category is pretrian- PA gulated in the sense of Bondal and Kapranov, [5]. Therefore, the category Ho A is triangulated with the collection of distinguished triangles being isomorphic toP those of the form (2.2).

Proof. The proof of this is the same as that of Propositions 1 and 2 of [5]. ! 2.5. Homotopy equivalences. As described above, a degree 0 closed morphism φ between cohesive modules Ei =(Ei•, Ei), i =1, 2, over A is a homotopy equivalence if it induces an isomorphism in Ho . We want to give a simple cri- PA terion for φ to define such a homotopy equivalence. On the complex A(E1,E2) define a decreasing filtration by P F k j (E ,E )= φ j (E ,E ) φi = 0 for i < k PA 1 2 { ∈PA 1 2 | } where φi is defined as in (2.1). Proposition 2.8. There is a spectral sequence pq p+q (2.3) E = H •(E ,E ) 0 ⇒ PA 1 2 where ! " pq p p+q +q p E0 = gr A•(E1,E2)= φ (E1,E2): E1• E2• P { ∈PA → ⊗A A } with differential p 0 p p+q p 0 d0(φ )=E φ ( 1) φ E 2 ◦ − − ◦ 1 DUALITY AND EQUIVALENCE 7

0 Proposition 2.9. A closed morphism φ A(E1,E2) is a homotopy equiva- 0 0 0 ∈P lence if and only if φ :(E1•, E1) (E2•, E2) is a quasi-isomorphism of complexes of -modules. → A Proof. Let E =(E•, E) be any object in A. Then φ induces a map of complexes P

(2.4) φ: •(E,E ) •(E,E ) PA 1 →PA 2

We show that the induced map on E1-terms of the spectral sequences are isomor- 0 0 phisms. Indeed, the quasi-isomorphism of (E•, E ) (E•, E ) implies that they 1 1 → 2 2 are actually chain homotopy equivalent since E1• and E2• are projective, hence for each p that 0 p 0 p 0 φ I:(E1• , E1 I) (E2• , E2 I) ⊗ ⊗A A ⊗ → ⊗A A ⊗ is a quasi-isomorphism and then

0 pq +q p pq +q p gr(φ)=φ : E0 = Hom (E•,E1• ) E0 = Hom (E•,E2• ) ∼ A ⊗A A → ∼ A ⊗A A is a quasi-isomorphism after one last double complex argument since the modules E• are projective over . Thus (2.4) is a quasi-isomorphism for all E and this implies φ is an isomorphismA in Ho . PA The other direction follows easily. !

2.6. The dual of a cohesive module. We define a duality functor which will be of use in future sections. Let A =( •, d, c) be a curved dga. Its opposite is A A◦ =( ◦•, d, c) where ◦• is the graded algebra whose product is given by A − A a b a b =( 1)| || |ba ·◦ − We will not use the notation for the product any longer. We can now define the ·◦ category of left cohesive modules over A as A◦ . We define the duality dg functor P

∨ : PA →PA◦ by

E =(E•, E) E∨ =(E∨•, E∨) %→ k k where E∨ = Hom (E− , ) and for φ E∨• A A ∈ φ (E∨φ)(e)=d φ(e) ( 1)| |φ E(e) − −

It is straightforward that E∨ is indeed! cohesive" over A!◦. There" is a natural pairing between E and E∨. And moreover the connection was deﬁned so that the relation

φ E∨(φ),e +( 1)| | φ, E(e) = d φ, e ( ) − ( ) ( ) holds. Note that the complex of morphisms A•(E1,E2) between cohesive modules can be identiﬁed with P

(E2• • E1∨•, 1 1 E1∨ +1 d 1+E2 1 1) ⊗A A ⊗A ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 8 J. BLOCK

2.7. Functoriality. We now discuss a construction of functors between categories of the form . Given two curved dgas, A =( •,d ,c ) and A = PA 1 A1 1 1 2 ( 2•,d2,c2) a homomorphism from A1 to A2 is a pair (f,ω) where f : 1• 2• isA a morphism of graded algebras, ω 1 and they satisfy A →A ∈A2 (1) f(d1a1)=d2f(a1) + [ω, f(a1)] and 2 (2) f(c1)=c2 + d2ω + ω . Given a homomorphism of curved dgas (f,ω) we deﬁne a dg functor

f : A A ∗ P 1 →P 2 as follows. Given E =(E•, E) a cohesive module over A1, set f ∗(E) to be the cohesive module over A2

(E• 1 2, E2) ⊗A A where E2(e b)=E(e)b+e (d2b+ωb). One checks that E2 is still an E-connection and satisﬁes⊗ ⊗ 2 (E2) (e b)= (e b)c2. ⊗ − ⊗ This is a special case of the following construction. Consider the following data, X =(X•, X) where

(1) X• is a graded ﬁnitely generated projective right- -module, A2 (2) X: X• X• 2 2• is a Z-connection, → ⊗A A (3) 1• acts on the left of X• 2 2• satisfying A ⊗A A a (x b) = (a x) b · · · · and X a (x b) = da (x b)+a X(x b) · ⊗ · ⊗ · ⊗ for a 1•, x X• and! b 2•, " ∈A ∈ ∈A (4) X satisﬁes the following condition:

X X(x b)=c1 (x b) (x b) c2 ◦ ⊗ · ⊗ − ⊗ · on the complex X• 2 2•. ⊗A A Let us call such a pair X =(X•, X) an A1-A2-cohesive bimodule. Given an A1-A2-cohesive bimodule X =(X•, X), we can then deﬁne a dg- functor (see the next section for the deﬁnition)

X∗ : PA1 →PA2 by

X∗(E•, E) = (E• 1 X•, E2) ⊗A where E2(e x)=E(e) x + e X(x), where the denotes the action of • on ⊗ · ⊗ · A1 X• 1 2•. One easily checks that X∗(E) is an object of A2 . We will write E X ⊗A A P ⊗ for E2. Remark 2.10. (1) The previous case of a homomorphism between curved dgas occurs by setting X• = in degree 0. • acts by the morphism f and the A2 A1 Z-connection is X(a2)=d2(a2)+ω a2. · DUALITY AND EQUIVALENCE 9

(2) To give another example of an A1-A2-cohesive bimodule, consider a manifold M with two vector bundles with connection (E , ) and (E , ). Let c 1 ∇2 2 ∇2 i be the curvature of i. Set Ai =( i•,di,ci)=( •(M; End(Ei), Ad i). Then we deﬁne a cohesive bimodule∇ betweenA them by settingA ∇

X• =Γ M; Hom(E2,E1) in degree 0. X• has a Z-connection! "

X(φ)(e2)= 1 φ(e2) φ( 2e2) ∇ − ∇ 2 and maps X• X• 2 2•. Then (X!) (φ)=" c1 φ φ c2 as is required. This → ⊗A A · − · cohesive bimodule implements a dg-quasi-equivalence between A1 and A2 . (See the next section for the deﬁnition of a dg-quasi-equivalence.) P P

3. Modules over PA It will be important for us to work with modules over A and not just with the objects of itself. P PA 3.1. Modules over a dg-category. We ﬁrst collect some general deﬁnitions, see [18] for more details.

Definition 3.1. A functor F : 1 2 between two dg-categories is a dg- functor if the map on hom sets C →C (3.1) F : (x, y) (F x, F y) C1 →C2 is a chain map of complexes. A dg-functor F as above is a quasi-equivalence if the maps in (3.1) are quasi-isomorphisms and Ho(F ): Ho 1 Ho 2 is an equivalence of categories. C → C Given a dg-category , one can define the category of (right) dg-modules C over , Mod- . This consists of dg-functors from the opposite dg-category ◦ to the dg-categoryC C (k) of complexes over k. More explicitly, a right -module CM is an assignment, toC each x , a complex M(x) and chain maps forC any x, y ∈C ∈C (3.2) M(x) (y, x) M(y) ⊗C → satisfying the obvious associativity and unit conditions. A morphism f Mod- (M,N) between right -modules M and N is an assignment of a map of∈ complexesC C (3.3) f : M(x) N(x) x → for each object x compatible with the maps in (3.2). Such a map is called ∈C a quasi-isomorphism if fx in (3.3) is a quasi-isomorphism of complexes for each x . One can make modules over a dg-category into a dg-category itself. The morphisms∈C we have defined in Mod- are the degree 0 closed morphisms of this dg-category. The category of left modulesC -Mod is defined in an analogous way. The category Mod- has a model structureC used by Keller to define its derived category, [18,19]. The quasi-isomorphismsC in Mod- are those we just defined. The fibrations are the componentwise surjections andC the cofibrations are defined by the usual lifting property. Using this model structure we may form the homotopy category of Mod- , obtained by inverting all the quasi-isomorphisms in Mod- . This is what Keller callsC the derived category of , and we will denote it by D(Mod-C ). C C 10 J. BLOCK

There is the standard fully faithful Yoneda embedding Z0 Mod- where x h = ( ,x). C→ C ∈C%→ x C · Moreover, the Yoneda embedding induces a fully faithful functor Ho D(Mod- ) C→ C This is simply because for an object x , the module h is trivially cofibrant. ∈C x Definition 3.2. (1) A module M Mod- is called representable if it is ∈ C isomorphic in Mod- to an object of the form hx for some x . (2) A module M Mod-C is called quasi-representable if it is isomorphic∈C in D(Mod- ) to∈ an objectC of the form h for some x . C x ∈C Definition 3.3. Let M Mod- and N -Mod. Their tensor product is defined to be the complex ∈ C ∈C

α M N = cok M(c) (c&,c) N(c&) M(c) N(c) ⊗C ⊗C ⊗ −→ ⊗ )c,c" c + *∈C *∈C where for m M(c), φ (c&,c) and n N(c&) ∈ ∈C ∈ α(m φ n)=mφ n m φn ⊗ ⊗ ⊗ − ⊗ Bimodules are the main mechanism to construct functors between module categories over rings. They play the same role for modules over dg-categories. Definition 3.4. Let and D denote two dg-categories. A bimodule X D-Mod- is a dg-functor C ∈ C X : ◦ D (k) C ⊗ →C More explicitly, for objects c, c& and d, d& D there are maps of complexes ∈C ∈ D(d, d&) X(c, d) (c&,c) X(c&,d&) ⊗ ⊗C → satisfying the obvious conditions. Definition 3.5. For a bimodule X D-Mod- and d D, we get an object ∈ C ∈ Xd Mod- where Xd(c)=X(c, d). ∈ C Similarly, for c , we get an object ∈C cX D-Mod where cX(d)=X(c, d). ∈ Therefore, we may define for M Mod-D the complex ∈ M cX ⊗D c Furthermore the assignment c X defines a functor ◦ D-Mod and so c c %→ C → %→ M D X defines an object in Mod- . Thus D X defines a functor from Mod-D Mod-⊗ . Moreover, by deriving thisC functor,·⊗ we get a functor → C L M M X %→ ⊗D from D(Mod-D) D(Mod- ). → C Definition 3.6 (Keller, [18]). A bimodule X D-Mod- is called a quasi- functor if for all d D, the object Xd Mod- is∈ quasi-representable.C Such a bimodule therefore defines∈ a functor ∈ C Ho D Ho . → C DUALITY AND EQUIVALENCE 11

Toen [27] calls quasi-functors right quasi-representable bimodules and it is a deep theorem of his that they form the correct morphisms in the localization of the category of dg-categories by inverting dg-quasi-equivalences. 3.2. Construction and properties of modules over . We now define a PA class of modules over the curved dga A that will define modules over the dg-category . PA Definition 3.7. For a curved dga A =( •, d, c), we define a quasi-cohesive A module to be the data of X =(X•, X) where X• is a Z-graded right module X• over together with a Z-connection A X: X• • X• • ⊗A A → ⊗A A that satisfies the integrability condition that the relative curvature F (x)=X X(x)+x c =0 X ◦ · for all x X•. Thus, they differ from cohesive modules by having possibly infin- itely many∈ nonzero graded components as well as not being projective or finitely generated over . A Definition 3.8. To a quasi-cohesive A-module X =(X•, X) we associate the -module, denoted h˜ , by PA X h˜X (E)= φ: E• • X• • of degree k and φ(xa)=φ(x)a a • { ⊗A A → ⊗A A | ∀ ∈A } with differential defined in the standard way φ d(φ)(ex)=X φ(x) ( 1)| |φ E(x) − − ˜ for all E =(E•, E) A. We use! hX "because of! its similarity" to the Yoneda embedding h, but beware∈P that X is not an object in . However, in the same way PA as A is shown to be a dg-category, h˜X is shown to be a module over A. For two quasi-cohesiveP A-modules X and Y , and P

f : X• • Y • • ⊗A A → ⊗A A of degree 0 and satisfying fX = Yf, we get a morphism of A-modules P h˜ : h˜ h˜ f X → Y The point of a quasi-cohesive A-module X =(X•, X) is that the differential and morphisms decompose just the same as they do for cohesive modules. For example, k k k+1 k X = X where X : E• X•− and similarly for morphisms. k → ⊗A A Proposition, 3.9. Let X and Y be quasi-cohesive A-modules and f a mor- 0 0 0 phism. Suppose f :(X•, X ) (Y •, Y ) is a quasi-isomorphism of complexes. → Then h˜ is a quasi-isomorphism in Mod- . The converse is not true. f PA It will be important for us to have a criterion for when a quasi-cohesive A- module X induces a quasi-representable -module. PA Definition 3.10. Define a map φ: C D between -modules to be alge- → A braically -nuclear, [24], if there are finite sets of elements φk Hom (C, ) and y D, kA=1,...,N such that ∈ A A k ∈ φ(x)= y φ (x) k · k -k 12 J. BLOCK

Proposition 3.11 (See Quillen, [24, Proposition 1.1]). For C• a complex of -modules, the following are equivalent: A (1) C• is homotopy equivalent to a bounded complex of ﬁnitely generated projective -modules. A (2) For any other complex of -modules D•, the homomorphism A Hom (C•, ) D• Hom (C•,D•) A A ⊗A → A is a homotopy equivalence of complexes (over k). (3) The endomorphism 1C of C• is homotopic to an algebraically nuclear endomorphism.

Deﬁnition 3.12. Suppose A =( •, d, c) is a curved dga. Let X =(X•, X) be A 0 a quasi-cohesive module over A. Suppose there exist -linear morphisms h : X• 1 0 A → X•− of degree 1 and T : X• X• of degree 0 satisfying − → (1) T 0 is algebraically -nuclear, A (2) [X0,h0] = 1 T 0 − Then we will call X a quasi-ﬁnite quasi-cohesive module. Our criterion is the following.

Theorem 3.13. Suppose A =( •, d, c) is a curved dga. Let X =(X•, X) be A a quasi-cohesive module over A. Then there is an object E =(E•, E) A such ∈P that h˜X is quasi-isomorphic to hE; that is, h˜X is quasi-representable, under either of the two following conditions: (1) X is a quasi-finite quasi-cohesive module. 0 (2) • is flat over and there is a bounded complex (E•, E ) of finitely Agenerated projectiveA right -modules and an -linear quasi-isomorphism 0 0 0 A A e :(E•, E ) (X•, X ). → Proof. In either case (1) or (2) of the theorem, there exists a bounded complex 0 of finitely generated projective right -modules (E•, E ) and a quasi-isomorphism 0 0 0 A e :(E•, E ) (X•, X ). In case (1), X is quasi-finite-cohesive, and Proposi- tion 3.11 implies→ that e0 is in fact a homotopy equivalence. In case (2) it is simply the hypothesis. In particular, e0E0 X0e0 = 0. Now we construct a Z-connection term by term. − The Z-connection X on X• induces a connection k 0 k 0 1 H: H (X•, X ) H (X•, X ) → ⊗A A for each k. We use the quasi-morphism e0 to transport this connection to a con- k 0 nection on H (E•; E )

k 0 k 0 1 H (E•; E ) / H (E•, E ) ⊗A A (3.4) e0 e0 1 ⊗ k 0 H k 0 1 H (X•, X ) / H (X•, X ) ⊗A A The right vertical arrow above e0 1 is a quasi-isomorphism; in case (1) this is 0 ⊗ because e is a homotopy equivalence and in case (2) because • is ﬂat. The ﬁrst step is handled by the following lemma. A DUALITY AND EQUIVALENCE 13

Lemma 3.14. Given a bounded complex of ﬁnitely generated projective mod- 0 k 0 k 0 1 A ules (E•, E ) with connections H: H (E•; E ) H (E•, E ) , for each k, there exist connections → ⊗A A k k 1 H: E E → ⊗A A lifting H. That is, 0 0 .HE =(E 1)H ⊗ and the connection induced on the cohomology is H. . . Proof (of lemma). Since E• is a bounded complex of -modules it lives in some bounded range of degrees k [N,M]. Pick an arbitraryA connection on EM , . Consider the diagram with exact∈ rows ∇ j M M 0 E / H (E•, ) / 0 RR E RRR (3.5) RRR θ RRR H ∇ RRR j 1 R( M 1 ⊗ M 0 1 E / H (E•, E ) / 0. ⊗A A ⊗A A In the diagram, θ = H j (j 1) is easily checked to be -linear and j 1 is surjective by the right◦ exactness− ⊗ of◦∇ tensor product. By the projectivityA of EM⊗, θ lifts to θ˜: EM EM 1 → ⊗A A so that (j 1)θ˜ = θ. Set H = + θ˜. With H in place of , the diagram above commutes.⊗ ∇ ∇ M 1 Now choose on E − .any connection M. 1. But M 1 does not necessar- 0 0 ∇ − ∇ − 0 ily satisfy E M 1 = HE = 0. So we correct it as follows. Set µ = HE 0 ∇ − 0 − (E 1) M 1. Then µ is -linear. Furthermore, Im µ Im E 1; this is because ⊗ ∇ − A ⊂ M⊗1 M 1 1 HE Im E 1 since H lifts. H. So by projectivity it lifts to θ : E − E − . ∈ ⊗0 M 1 M 1 1 → ⊗A A such that (E 1) θ = θ. Set H: E − E − to be M 1 + θ. Then ⊗ ◦ → ⊗A A ∇ − E.0H = HE0 in the right. most square below. .

0. . 0 0 0 . EN E / EN+1 E / ... E / EM 1 E / EM / 0 . . − M MMM µ (3.6) M 1 MM ∇ − MMM H 0 1 0 1 0 1 0 1 M& N 1 E ⊗ N+1 1 E ⊗ E ⊗ M 1 1 E ⊗ Me 1 E / E / ... / E − / E / 0. ⊗AA ⊗AA ⊗AA ⊗AA 1 0 Now we continue backwards to construct all H: E• E• satisfying (E 1)H 0 → ⊗AA ⊗ = HE = 0. This completes the proof of the lemma. ! . . (Proof of the theorem, continued). Set E1 =( 1)kH on Ek. Then . − 0 1 1 0 E E + E E =0 . . but it is not necessarily true that e0E1 X1e0 = 0. We correct this as follows. 0 1 1 0 . .− 1 Consider ψ = e E X e : E• X• . Check that ψ is -linear and a map of complexes. − → .⊗A A A

. 1 0 (E• , E 1) ⊗6 A A ⊗ ψ˜ mm mm 0 (3.7) mmm e 1 mmm ⊗ mmψ mm 1 0 E• / (X• , X 1). ⊗A A ⊗ 14 J. BLOCK

In the above diagram, e0 1 is a quasi-isomorphism since e0 is a homotopy equiv- ⊗ 1 alence. So by [20, Lemma 1.2.5] there is a lift ψ˜ of ψ and a homotopy e : E• 1 1 0 → X•− between (e 1)ψ˜ and ψ, ⊗A A ⊗ 0 1 0 0 1 ψ (e 1)ψ =(e E + X e ) − ⊗ So let E1 = E1 ψ. Then − . 0 1 1 0 0 1 1 0 1 0 0 1 (3.8) E E + E E = 0 and e E X e = e E + X e . . . − So we have constructed the ﬁrst two components E0 and E1 of the Z-connection 0 1 and the ﬁrst components e and e of the quasi-isomorphism E• • X• •. 0 ⊗AA → ⊗AA To construct the rest, consider the mapping cone L• of e . Thus

L• = E[1]• X• ⊕ Let L0 be deﬁned as the matrix 0 0 E [1] 0 (3.9) L = 0 0 e [1] X ' ( Deﬁne L1 as the matrix 1 1 E [1] 0 (3.10) L = 1 1 e [1] X ' ( Now L0L0 = 0 and [L0, L1] = 0 express the identities (3.8). Let

1 1 00 (3.11) D = L L + 2 0 0 2 + rc X e [X , X ] ' ( where rc denotes right multiplication by c. Then, as is easily checked, D is -linear and A (1) [L0,D] = 0 and (2) D 0 X = 0. ⊕ • |0 Since (L•, L ) is the mapping cone of a quasi-isomorphism, it is acyclic and since 2 0 • is ﬂat over ,(L• , L 1) is acyclic too. Since E• is projective, we have thatA A ⊗A A ⊗ 0 2 0 Hom• (E•, E ), (L• , L ) A ⊗A A is acyclic. Moreover ! " 0 2 0 2 0 Hom• (E•, E ), (L• , L ) Hom• L•, (L• , [L , ]) A ⊗A A ⊂ A ⊗A A · 2 is a subcomplex.! Now we have that D " Hom• (E! •,L• ) is a cycle" and so 2 2 ∈ A 0 ⊗A2 A 2 there is L Hom• (E•,L• ) such that D =[L , L ]. Deﬁne L on L• by ∈ A ⊗A A − 2 2 00 (3.12) . L = L + 2 . . 0 X ' ( Then .

0 2 0 2 00 0 2 00 1 1 (3.13) [L , L ]= L , L + 2 = D+ L , L + 2 = L L rc. 0 X − 0 X − − / ' (0 / ' (0 So . . 0 2 1 1 2 0 L L + L L + L L + rc =0. We continue by setting

1 2 2 1 00 (3.14) D = L L + L L + 3 0 0 3 . X e [X , X ] ' ( DUALITY AND EQUIVALENCE 15

3 Then D : L• L• is -linear, D 0 X = 0 and → ⊗A A A | ⊕ • 0 1 1 [L ,D]=L rc rc L =0 ◦ − ◦ by the Bianchi identity d(c) = 0. Hence, by the same reasoning as above, there is 3 3 0 3 L Hom• (E•,L• ) such that D =[L , L ]. Define ∈ A ⊗A A − 3 3 00 (3.15). L = L + 3. . 0 X ' ( 3 i. 3 i Then one can compute that i=0 L L − = 0. Now suppose we have defined L0,..., Ln satisfying for k =0, 1, . . . , n , k i k i L L − = 0 for k =2 - i=0 - and 2 i 2 i L L − + rc = 0 for k = 2. i=0 - Then define n 00 (3.16) D = i n+1 i + . L L − n+1e0 [ 0, n+1] i=1 X X X - ' ( D 0 X = 0 and we may continue the inductive construction of L to finally arrive | ⊕ • at a Z-connection satisfying LL + rc = 0. The components of L construct both the Z-connection on E• as well as the morphism from (E•, E) to (X•, X). !

4. Complex manifolds We justify our framework in this section by showing that, for a complex manifold, the derived category of sheaves on X with coherent cohomology is equivalent to the homotopy category A for the Dolbeault algebra. Throughout this section P 0, let X be a compact complex manifold and A =( •, d, 0) = ( •(X), ∂¯, 0) the Dol- A A 0, beault dga. This is the global sections of the sheaf of dgas ( • , d, 0) = ( •, ∂¯, 0). AX AX Let X denote the sheaf of holomorphic functions on X. Koszul and Malgrange haveO shown that a holomorphic vector bundle ξ on a complex manifold X is the same thing as a C∞ vector bundle with a ﬂat ∂¯-connection, i.e., an operator 1 ∂¯ξ : Eξ Eξ → ⊗A A such that ∂¯ (fφ)=∂¯(f)φ + f∂¯ (φ) for f , φ Γ(X; ξ) and satisfying the ξ ξ ∈A ∈ integrability condition that ∂¯ξ ∂¯ξ = 0. Here Eξ denotes the global C∞ sections of ξ. The notion of a cohesive◦ module over A clearly generalizes this notion but in fact will also include coherent analytic sheaves on X and even more generally, bounded complexes of -modules with coherent cohomology as well. OX For example, if (ξ•,δ) denotes a complex of holomorphic vector bundles, with i i 1 corresponding global C∞-sections E• and ∂¯-operator ∂¯ξ : E E then the ¯ ¯ → ⊗A A 0 holomorphic condition on δ is that δ∂ξ = ∂ξδ. Thus E =(E•, E), where E = δ 1 ¯ and E =( 1)•∂ξ deﬁnes the cohesive module corresponding to (ξ•,δ). So we see that, for coherent− sheaves with locally free resolutions, there is nothing new here. 16 J. BLOCK

4.1. The derived category of sheaves of -modules with coherent OX cohomology. Pali, [22] was the first to give a characterization of general coherent ¯ analytic sheaves in terms of sheaves over ( X• ,d) equipped with flat ∂-connections. He defines a ∂¯-coherent analytic sheaf toA be a sheaf of modules over the sheaf of F C∞-functions X satisfying two conditions: Finiteness:A locally on X, has a finite resolution by finitely generated free modules, and F Holomorphic: is equipped with a ∂¯-connection, i.e., an operator (at the level of sheaves) F ¯ 1 ∂ : X X F→F⊗A A and satisfying ∂¯2 = 0. Theorem 4.1 (Pali, [22]). The category of coherent analytic sheaves on X is equivalent to the category of ∂¯-coherent sheaves. We prove our theorem independently of his proof. We use the following proposition of Illusie, [3].

Proposition 4.2. Suppose (X, X ) is a ringed space, where X is compact and is a soft sheaf of rings. Then A AX (1) The global sections functor Γ: Mod- Mod- (X) AX → AX is exact and establishes an equivalence of categories between the category of sheaves of right X -modules and the category of right modules over the global sections (X). A AX (2) If M Mod- X locally has finite resolutions by finitely generated free -modules, then∈ Γ(XA; M) has a finite resolution by finitely generated projectives. AX (3) The derived category of perfect complexes of sheaves Dperf (Mod- X ) is equivalent to the derived category of perfect complexes of modulesA D Mod- (X) . perf AX Proof.! See [3," Proposition 2.3.2, ExposéII]. ! Our goal is to derive the following description of the bounded derived category of sheaves of X -modules with coherent cohomology on a complex manifold. Note that this is equivalentO to the category of perfect complexes, since we are on a smooth 0, manifold. Recall that A =( •, d, 0) = ( •(X), ∂¯, 0), the Dolbeault dga, is the A A 0, global sections of the sheaf of dgas ( • , d, 0) = ( •, ∂¯, 0) AX AX Theorem 4.3. Let X be a compact complex manifold and A =( •, d, 0) = 0, A ( •(X), ∂¯, 0) the Dolbeault dga. Then the category Ho is equivalent to the A PA bounded derived category of complexes of sheaves of X -modules with coherent co- b O homology Dcoh(X). Remark 4.4. This theorem is stated only for X compact. This is because Proposition 4.2 is stated only for X compact. A version of Theorem 4.3 will be true once one is able to characterize the perfect X -modules in terms of modules over the global sections for X which are not compact.A A module M over naturally localizes to a sheaf M of -modules, where A X AX MX (U)=M X (U) ⊗A A DUALITY AND EQUIVALENCE 17

p,q For an object E =(E•, E) of A, deﬁne the sheaves by P EX p,q(U)=Ep q (U). EX ⊗A AX p,q We deﬁne a complex of sheaves by ( X• , E) = ( p+q= X , E). This is a complex E • E of soft sheaves of -modules, since is a ∂¯-connection. The theorem above will X E , be broken up intoO several lemmas.

Lemma 4.5. The complex • has coherent cohomology and EX E =(E•, E) α(E) = ( • , E) %→ EX deﬁnes a fully faithful functor α: Ho (X) b (X). PA →Dperf 2Dcoh Proof. Let U be a polydisc in X. We show that on a possibly smaller polydisc V , there is gauge transformation φ: • V • V of degree zero such that 1 0 ¯ E | →E | φ E φ− = F + ∂. Thus • V is gauge equivalent to a complex of holomorphic ◦ ◦ E | p ,0 0 ¯ vector bundles. Or, in other words, for each p the sheaf H • , E is ∂-coherent, E with ∂¯-connection 1. Since U is Stein there is no higher cohomology (with re- E ! " spect to E1) and we are left with the holomorphic sections over U of each of these ∂¯-coherent sheaves, which are thus coherent. The construction of the gauge transformation follows the proof of the integrability theorem for complex structures on vector bundles, [9, Section 2.2.2, p. 50]. Thus we may assume we are in a polydisc U = (z , . . . , z ) z

(3) ι∂/∂z¯1 J1 = 0. We now iterate this procedure. Write J = J & dz¯ +J && where ι J & = ι J & = 1 1∧ 2 1 ∂/∂z¯1 1 ∂/∂z¯2 1 ι∂/∂z¯1 J1&& = ι∂/∂z¯2 J1&& = 0. Now solve 1 φ− ∂¯ (φ )=J & dz¯ 2 2 2 1 ∧ 2 18 J. BLOCK for φ2. Since J1& is holomorphic in z1 and smooth in z2, . . . , zn, so will φ2. Then as before we have 1 φ (∂¯ + J & dz¯ )φ− = ∂¯ 2 2 1 ∧ 2 2 2 as well as ¯ 1 ¯ φ2(∂1)φ2− = ∂1 1 since φ2 is holomorphic in z1. Setting E2 = φ2 E1 φ− , we see that ◦ ◦ 2 0 ¯ ¯ ¯ E2 = E2 + ∂1 + ∂2 + ∂ 3 + J2 ≥ 0 where ι∂/∂z¯1 J2 = ι∂/∂z¯2 J2 = 0 and we can check as before that both E2 and J2 are 0 ¯ holomorphic in z1 and z2. We continue until we arrive at F = En = En + ∂. !

Lemma 4.6. To any complex of sheaves of -modules ( • ,d) on X with OX EX coherent cohomology there corresponds a cohesive A-module E =(E•, E), unique up to quasi-isomorphism in and a quasi-isomorphism PA α(E) ( •,d) → E This correspondence has the property that, for any two such complexes • and •, E1 E2 the corresponding twisted complexes (E1•, E1) and (E2•, E2) satisfy k k Ext ( 1•, 2•) = H A(E1,E2) OX E E ∼ P Proof. Since we are on a manifold we! may assume" that ( •,d) is a perfect E ¯ complex. Set • = • X X . Now the map ( •,d) ( • X• ,d 1+1 ∂) is a quasi-isomorphismE∞ E ⊗ ofO sheavesA of -modulesE by the→ E flatness∞ ⊗A A of ⊗ over ⊗ . OX AX OX Again, by the flatness of X over X , it follows that ( • ,d) is a perfect com- A O E∞ plex of X -modules. By Proposition 4.2, there is a (strictly) perfect complex 0 A 0 0 (E•, E ) of -modules and quasi-isomorphism e :(E•, E ) (Γ(X, • ),d). More- A → E∞ over (Γ(X, • ),d 1 + 1 ∂¯) defines a quasi-cohesive module over A. So the E∞ ⊗ ⊗ hypotheses of Theorem 3.13(2) are satisfied. The lemma is proved. ! 4.2. Gerbes on complex manifolds. The theorem above has an analogue for gerbes over compact manifolds. X is still a compact complex manifold. A 2 class b H (X, X× ) defines an X× -gerbe on X. From the exponential sequence of sheaves∈ O O exp 2πı 0 ZX X · × 0 → →O −−−−−→OX → there is a long exact sequence 2 2 3 H (X; X ) H (X; × ) H (X; ZX ) ···→ O → OX → →··· If b maps to 0 H3(X; Z) (that is, the gerbe is topologically trivializable) then b pulls back to a∈ class represented by a (0, 2)-form B 0,2(X). Consider the curved 0, ∈A dga A =( •, d, B)=( •(X), ∂¯,B) — the same Dolbeault algebra as before but with a curvature.A ThenA we have a theorem [4], corresponding to (4.3), Theorem 4.7. The category Ho is equivalent to the bounded derived cate- PA gory of complexes of sheaves on the gerbe b over X of X -modules with coherent b O cohomology and weight one Dcoh(X)(1). Sheaves on a gerbe are often called twisted sheaves. One can deal with gerbes which are not necessarily topologically trivial, but the curved dga is slightly more complicated, [4]. DUALITY AND EQUIVALENCE 19

5. Examples 5.1. Elliptic curved dgas. In this section we define a class of curved dga’s A such that the corresponding dg-category A is proper, that is, the cohomology of the hom sets are finite dimensional. It isP often useful to equip a manifold with a Riemannian metric so that one can use Hilbert space methods. We introduce a relative of the notion spectral triple in the sense of Connes, [6], so that we can use Hilbert space methods to guarantee the properness of the dg-category. Again, our basic data is a curved dga A =( •, d, c). A Definition 5.1. We say that A is equipped with a Hilbert structure if there is a positive definite Hermitian inner product on • A k k , : C (· ·) A ×A → satisfying the following conditions: Let • be the completion of •. H A (1) For a •, the operator l (respectively r ) of left (respectively, right) ∈A a a multiplication by a extends to • as a bounded operator. Furthermore, H the operators la∗ and ra∗ map • • to itself. (2) has an anti-linear involutionA ⊂: H such that for a , there is A ∗ A→A ∈A (la)∗ = la∗ and (ra)∗ = ra∗ . (3) The differential d is required to be closable in •. Its adjoint satisfies H d∗( •) • and the operator D = d + d∗ is essentially self-adjoint with A ⊂A core •. A (4) For a •,[D, l ], [D, r ], [D, l∗] and [D, r∗] are bounded operators ∈A a a a a on •. H Definition 5.2. An elliptic curved dga A =( •, d, c) is a curved dga with a Hilbert structure which in addition satisfies A tD2 (1) The operator e− is trace class for all t>0. n (2) • = Dom(D ) A n The following2 proposition follows from very standard arguments.

Proposition 5.3. Given an elliptic curved dga A then for E =(E•, E) and F =(F •, F) in A one has that the cohomology of A(E,F ) is finite dimensional. P P Bondal and Kapranov have given a very beautiful formulation of Serre duality purely in the derived category. We adapt their definitions to our situation of dg- categories. Definition 5.4. For a dg-category , such that all Hom complexes have finite dimensional cohomology, a Serre functorC is a dg-functor S : C→C which is a dg-equivalence and so that there are pairings of degree zero, functorial in both E and F , : •(E,F ) •(F,SE) C[0] (· ·) C ×C → satisfying φ dφ,ψ +( 1)| | φ,dψ =0 ( ) − ( ) which are perfect on cohomology for any E and F in . C Motivated by the case of Lie algebroids below, we make the following definition, which will guarantee the existence of a Serre functor. 20 J. BLOCK

Deﬁnition 5.5. Let A =( •, d, c) be an elliptic curved dga. A dualizing A module (of dimension g) is a triple ((D, D), ¯, ) where ∗ (1) (D, D) is an A-A cohesive bimodule, k g k 3 (2) ¯: D − is a conjugate linear isomorphism and satisﬁes ∗ A → ⊗A A ¯(aω)=¯(ω)a∗ and ¯(ωa)=a∗¯(ω) ∗ ∗ ∗ ∗ for a and ω •. ∈A ∈A (3) There is a C-linear map : D g C such that D(x) = 0 for all ⊗A A → x D • and ∈ ⊗A A 3 3 ω x ω x =( 1)| || | x ω · − · 4 4 for all ω • and x D •, and ∈A ∈ ⊗A A ω,η = ¯(ω)η ( ) ∗ 4 Proposition 5.6. Given an elliptic curved dga A =( •, d, c) with a dualizing A module ((D, D), ¯, ), the category A has a Serre functor given by the cohesive ∗ P bimodule (D[g], D). That is, 3 S(E•, E) = (E D[g], E # D) ⊗A is a dg-equivalence for which there are functorial pairings

, : •(E,F ) •(F,SE) C (· ·) PA ×PA → satisfying φ dφ,ψ +( 1)| | φ,dψ =0 ( ) − ( ) is perfect on cohomology for any E and F in . PA 5.2. Lie algebroids. Lie algebroids provide a natural source of dga’s and thus, by passing to their cohesive modules, interesting dg-categories. Let X be a C∞-manifold and let a be a complex Lie algebroid over X. Thus a is a C∞ vector bundle on X with a bracket operation on Γ(X; a) making Γ(X; a) into a Lie algebra and such that the induced map into vector ﬁelds ρ:Γ(X; a) (X) →V is a Lie algebra homomorphism and for f C∞(X) and x, y Γ(X; a) we have ∈ ∈ [x, fy]=f[x, y] + (ρ(x)f)y. Let g be the rank of a and n for the dimension of X. There is a dga corresponding to any Lie algebroid a over X as follows. Let

• =Γ(X; •a∨) Aa denote the space of smooth a-diﬀerential forms.5 It has a diﬀerential d of degree one, with d = 0 given by the usual formula,

(5.1) (dη)(x , . . . , x )= ( 1)i+1ρ(x ) η(x ,..., xˆ , . . . , x ) 1 k − i 1 i k i - ! " + ( 1)i+jη([x ,x ],..., xˆ ,..., xˆ , . . . , x ). − i j i j k i

5.2.1. The dualizing a-module a. We recall the definition of the “dualizing module” of a Lie algebroid. This wasD first defined in [12] where they used it to define the modular class of the Lie algebroid. Let a be a Lie algebroid over X with anchor map ρ. Consider the line bundle g n (5.2) = a T ∨X. Da ⊗ C Write D =Γ(X; ). Define a Da 5 5 1 D: Da Da a a → ⊗A A by

(5.3) D(X µ)(x)=Lx(X) µ + X Lρ(x)µ ⊗ ⊗ ⊗ where x Γ(X; a), X Γ(X; ga), µ Γ( nT X), and L µ denotes the Lie C∨ ρ(x) derivative∈ of µ in the direction∈ of ρ(x).∈ See [12] for more details. 5 5 1 Now we note that Aa acts on the left of Da a a• and D: Da Da a a ⊗A A → ⊗A A defines a flat •-connection [12]. Therefore (Da, D) denote a cohesive Aa-Aa- Aa bimodule, and thus a dg-functor from Aa to itself. We have the pairing P ga n a ∨ TC∗X. D ⊗ g → g Which allows us to define : Da C for (X µ) ν a a∨ ⊗A5A → 5 ⊗ ⊗ ∈D ⊗ 3 (X,ν)µ 5 4 Then we have g Theorem 5.7 (Stokes’ theorem, [12]). Identify Da a a(X)= g g n ⊗A A Γ( a a∨ T ∨X) with the space of top-degree forms on X by pairing ⊗ ⊗g C g the factors in a∗ and a pointwise. We have, for every c =(X µ) ν 5 5 g 1 5 ⊗ ⊗ ∈ D a− (X), Aa ⊗Aa A 5 5 g 1 (5.4) D(c) = ( 1) − d(ρ(µ X) ν). − " " Consequently,

(5.5) D(c) = 0. 4X 5.2.2. Hermitian structures and the ¯-operator. Let us equip the algebroid a ∗ with a Hermitian inner product , . Then a∨ and •a∨ all inherit Hermitian inner products according to the rule( ) 5 α α ,β β = det( α ,β ). ( 1 ∧···∧ k 1 ∧···∧ k) ( i j) Also let us put on X a Riemannian structure and let νX be the volume form. Then there is a Hermitian inner product on •(X) defined by Aa α,β = α,β ν . ( ) ( ) X 4X Recall that there is a canonical identification of ga with nT X. Define a ∨ C∨ k g k D ⊗ the operator ¯: a∨ − a∨ by requiring that ∗ →Da ⊗ 5 5 (5.6) α ¯β = α,β ν . 5 5 ∧ ∗ ( ) X This is well defined because the pairing k g k n a∨ ( − a∨) T ∨X × Da ⊗ → C 5 5 5 22 J. BLOCK is perfect. Our ¯ operator is conjugate linear. This is because we have no conjugation operator on∗ a, as would be the case when we define the Hodge operator on the bigraded Dolbeault complex. ∗ As usual, we have the familiar local expressions for the ¯-operator. So if 1 g ∗ α1,...,αg is an orthonormal frame of a with α ,...,α the dual frame, then for a multi-index I 1, . . . , g we have ⊂{ } I σ(I) ¯ Ic ¯(λα ) = ( 1) (α 1,...,g νX ) λα ∗ − { } ⊗ ⊗ where Ic is the complement of the multi-index and σ(I) is the sign of the permuta- c tion (1, . . . , g) (I,I ). For an object (E,E) of Aa we equip E with a Hermitian structure (no condition)%→ and we extend ¯ to P ∗ k g k ¯: E a∨ E∨ − a∨ ∗ ⊗ → ⊗Da ⊗ by the same formula, (5.6). Locally,5 we have 5

I σ(I) i ¯ Ic ¯(ei λα ) = ( 1) e (α 1,...,g νX ) λα ∗ ⊗ − ⊗ { } ⊗ ⊗ i where ei and e are dual pairs of orthonormal frames of E and E∨ respectively. Now we make a basic assumption on our Lie algebroid a.

Deﬁnition 5.8. A complex Lie algebroid ρ: a T X is called elliptic if → C

ρ∨ : T ∨X T ∨X a∨ → C → is injective.

Note that a real Lie algebroid being elliptic means that it is transitive. The point of this deﬁnition is the following proposition.

Theorem 5.9. For an elliptic Lie algebroid a, the corresponding dga Aa = ( a•, d, 0) is an elliptic dga and ((Da, D), ¯, ) is a dualizing manifold with a repre- sentation.A ∗ 3 Proof. Everything follows from basic elliptic theory. !

As an immediate corollary we have

Theorem 5.10. For an elliptic Lie algebroid a with (E,E), (F, F) Aa , there is a perfect duality pairing ∈P

k g k H Aa (E,F ) H − Aa (F,E Da) C. P × P ⊗ → 5.2.3. The de Rham! Lie algebroid" and! Poincaréduality." For ρ = Id: a = TM TM the duality theorem is Poincaré’sfor local systems. That is, the dualizing→ module is the trivial one-dimensional vector bundle (we made the blanket assumption that M is orientable) and for a flat vector bundle E over X there is a perfect pairing k n k H (X; E) H − (X,E∨) C. × → DUALITY AND EQUIVALENCE 23

5.2.4. The Dolbeault Lie algebroid and Serre duality. For X a complex n- 0,1 dimensional manifold, let ρ: a = T 1 TCX be the natural inclusion. Thus, a holomorphic vector bundle is the same→ thing as an T 0,1-module. Moreover n 0,1 2n n 0,1 n 0,1 1,0 n 1,0 0,1 = T T ∨X = T (T ∨X T ∨X) = T ∨X DT ⊗ C ∼ ⊗ ⊕ ∼ is the usual5 canonical5 (or dualizing)5 bundle5 K in complex geometry.5 And Theo- rem 5.10 reduces to Serre’s duality theorem that for a holomorphic vector bundle E the sheaf (i.e., Dolbeault) cohomology satisﬁes k n k H (X; E)∨ = H − (X; E∨ K). ∂¯ ∼ ∂¯ ⊗ p from which it follows by letting E be T 1,0∨ that p,q n p,n q H (X) ∼=5H − − (X). Stated in terms of Serre functors, we have that with SE = E K[n], S is a Serre ⊗ functor on T 0,1 . 5.2.5. PThe Higgs Lie algebroid. Again, let X be an n-dimensional complex manifold. We deﬁne a new Lie algebroid as follows.

p"" a = T X = T 0,1 T 1,0 T X C ⊕ −→ C 0,1 where p&& is the projection of the complexiﬁed tangent bundle onto T X. Let p& be the projection onto T 1,0X. We need to adjust the bracket by

X& + X&&,Y& + Y && =[X&&,Y&&]+p&([X&&,Y&]+[X&,Y&&]). { } 1,0 0,1 for X&,Y& Γ(T ) and X&&,Y&& Γ(T ) and where the square brackets denote the usual bracket∈ of vector fields. ∈ Proposition 5.11. (1) a is an elliptic Lie algebroid. (2) A module over a is comprised of the following data:(E,Φ) where E is a 1,0 holomorphic vector bundle and Φ is a holomorphic section of Hom(E,E T ∨X) and satisfies the integrability condition Φ Φ= 0, that is,(E,Φ) is a Higgs⊗ bundle in the sense of Hitchin [14], and Simpson∧[26]. (3) The dualizing module a is the trivial one-dimensional vector bundle with the Higgs field Φ= 0. D Proof. That , satisfies Jacobi is a straightforward calculation that only uses the integrability{· of·} the complex structure, that is, that T 0,1 and T 1,0 are both closed under bracket. To check the algebroid condition we calculate (5.7) X, fY { } = X& + X&&, fY & + fY && =[X&& + fY &&]+p&([X&, fY &&]+[X&&, fY &]) { } = f[X&&,Y&&]+X&&(f)Y && + p&(f[X&,Y&&]+X&(f)Y && + f[X&&,Y&]+X&&(f)Y &)

= f [X&&,Y&&]+p&([X&,Y&&]+[X&&,Y&]) + X&&(f)(Y & + Y &&)+p&(X&(f)Y &&)

= f X,Y + p&&(X)(f)Y. {! } " To show it is an elliptic Lie algebroid, let ξ TX . Since its image in T X is real ∨ C∨ 0,1 ∈ 0,1 it can be written as e +¯e for e T ∨. The projection to T ∨ is simply e and ∈ thus ρ∨ is injective from TX∨ a∨. → For the statement about modules, suppose (E,E) is a module over a. Then we have the decomposition 1,0 0,1 1,0 0,1 E:Γ(E) Γ E (T ∨ T ∨) = Γ(E T ∨) Γ(E T ∨) → ⊗ ⊕ ∼ ⊗ ⊕ ⊗ ! " 24 J. BLOCK in which E decomposes as E = E& E&&. The condition of being an a connection ⊕ ¯ means that E&& satisfies Leibniz with respect to the ∂-operator and E& is linear over 1,0 2 the functions and thus Φ= E& : E E T ∨. The flatness condition =0 implies → ⊗ ∇ 2 (1) E&& = 0 and thus defines a holomorphic structure on E, (2) E&& Φ+Φ E&& = 0 and so Φis a holomorphic section, (3) Φ ◦Φ= 0. ◦ ∧ The statement about the dualizing module is also clear. The duality theorem in this case is due to Simpson, [26]. ! 5.2.6. Generalized Higgs algebroids. The example above is a special case of a general construction. Let ρ: a TCX be a Lie algebroid and (E,E) a module over → ρ a. Then set a = a E with the anchor map being the composition a E a E ⊕ ⊕ → −→ TCX. Define the bracket as

[X1 + e1,X2 + e2]E =[X,Y ]+EX1 e2 EX2 e1. − Proposition 5.12. (1) aE is a Lie algebroid. (2) If a is elliptic, then aE is elliptic as well. (3) A module (H, H = H0 +Φ) over aE consists of a triple (H, H0, Φ) where (H, H0) is an a-module and Φ: H H a∨ satisﬁes [H0, Φ] = 0 (i.e., Φ is a morphism of a-modules) and Φ→ Φ⊗= 0. ∧ Proof. All of these statements follow as in the previous example. !

We call such a triple (H, H0, Φ) a Higgs bundle with coeﬃcients in E. 5.2.7. The generalized complex Lie algebroid. Recall from [15] and [13] that an almost generalized complex structure on a manifold X is deﬁned by a subbundle

E (TX T ∨X) ⊆ ⊕ C satisfying that E is a maximal isotropic complex subbundle E (TX TX∨) ⊂ ⊕ C such that E E = 0 . The isotropic condition is with respect to the bilinear form ∩ { } X + ξ,Y + η = 1 ξ(Y )+η(x) ( ) 2 The almost generalized complex structure E! is integrable" and E is called a generalized complex structure if the sections of E,Γ(E), are closed under the Courant bracket. The Courant bracket is a skew-symmetric bracket deﬁned on smooth sections of (TX TX∨) , given by ⊕ C [X + ξ,Y + η] = [X,Y ]+L η L ξ 1 d(i η i ξ), X − Y − 2 X − Y where X + ξ,Y + η Γ(TX TX∨)C. It is shown in [15] and [13] how symplectic and complex manifolds∈ are examples⊕ of generalized complex manifolds.

In the case of a generalized complex structure, the projection map ρ: E TCX deﬁnes a Lie algebroid, the Lie algebra structure on the sections of E being→ the Courant bracket. Note that the Courant bracket on the full space (TX TX∨) C does not satisfy Jacobi. ⊕ ⊗ Proposition 5.13. E is an elliptic Lie algebroid. Proof. That it is a Lie algebroid is a straightforward calculation, as in [13]. That it is elliptic follows just as in the case of the Higgs Lie algebroid. ! DUALITY AND EQUIVALENCE 25

In this case, Gualitieri [13] calls cohesive modules generalized holomorphic vector bundles. There is therefore a duality theorem in this context. In general it can- not be made any more explicit than the general duality theorem (Theorem 5.10). On the other hand, in the special case where the generalized complex manifold is 0,1 1,0 a complex manifold X, E = T X T ∨X, [13]. Then we have ⊕ Proposition 5.14. Let E be the algebroid coming from the generalized complex structure defined by an honest complex structure as defined above. Then (1) a module over E consists of the following data:(E,Φ) where E is a holomorphic vector bundle on X, and Φ Hom(E,E T 1,0) is a holomorphic section and satisfies Φ Φ= 0. ∈ ⊗ ∧ 2 (2) The dualizing module E is (K⊗ , 0), the square of the canonical bundle with the zero Higgs fieldD 0.

Proof. The proof is the same as for the Higgs algebroid. ! 5.3. Noncommutative tori. 5.3.1. Real noncommutative tori. We now describe noncommutative tori. We will describe them in terms of twisted group algebras. Let V be a real vector space, and Λ V a lattice subgroup. The we can form the group ring S ∗(Λ), the Schwartz space⊂ of complex valued functions on Λwhich decrease faster than any 2 polynomial. Let B Λ V ∨, and form the biadditive, antisymmetric group cocycle σ :Λ Λ U(1) by∈ × → 2πıB(λ1,λ2) σ(λ1,λ2)=e In our computations, we will often implicitly make use of the fact that σ is biadditive and antisymmetric. Now we can form the twisted group algebra (Λ; σ) consisting A of the same space of functions as S ∗(Λ) but where the multiplication is defined by [λ ] [λ ]=σ(λ ,λ )[λ + λ ] 1 ◦ 2 1 2 1 2 1 This is a -algebra where f ∗(λ)=f(λ ). This is one of the standard ways to ∗ − describe the (smooth version) of the noncommutative torus. Given ξ V ∨ it is easy to check that ∈ (5.8) ξ(f)(λ) = 2πı ξ,λ f(λ) ( ) defines a derivation on (Λ; σ). Note that the derivation ξ is “real” in the sense A that ξ(f ∗)= ξ(f). Finally define a (de Rham) dga A by − •(Λ; σ)= (Λ; σ) Λ•V A A ⊗ C where V = V C and the differential d is defined on functions φ (Λ; σ) by C ⊗ ∈A df, ξ = ξ(f) ( ) for ξ V . In other words, for λ Λone has dλ =2πıλ D(λ) where D(λ) C∨ ∈ 1 ∈ ⊗ denotes λ as an element of Λ V . Extend d to the rest of •(Λ; σ) by Leibniz. Note that d2 = 0. A Remark 5.15. We just want to point out that V appears as the “cotangent” space. This is a manifestation of the fact that there is a duality going on. That is, in the case that σ = 1, we have that the dga A =( •(Λ; σ), d, 0) is naturally isomor- A phic to ( •(V ∨/Λ∨), d, 0), the de Rham algebra of the dual torus, and T0∨(V ∨/Λ∨) is naturallyA isomorphic to V . See Proposition 5.16 for the complex version of this. 26 J. BLOCK

5.3.2. Complex noncommutative tori. We are most interested in the case where our torus has a complex structure and in defining the analogue of the Dolbeault DGA for a noncommutative complex torus. So now let V will be a vector space with a complex structure J : V V , J 2 = . Let g be the complex dimension → − of V . Set VC = V R C. Then J 1: VC VC still squares to and so VC decomposes into ı and⊗ ı eigenspaces,⊗ V →V . The dual V also− decomposes 1,0 0,1 C∨ as V = V 1,0 V 0,1. Let− D : V V⊕ and D : V V denote the C∨ & C C 1,0 && C 0,1 corresponding⊕ projections. Explicitly⊗ → ⊗ → J 1+1 ı D& = ⊗ ⊗ 2ı and J 1+1 ı D&& = − ⊗ ⊗ 2ı and D = D& + D&& where D denotes the identity. This also establishes a decomposition k p,q Λ VC = Λ V p+q= 6• p,q p q where Λ V =Λ V1,0 Λ V0,1. Complex conjugation on VC defines an involution and identifies V with the⊗ v V such thatv ¯ = v. ∈ C Now let X = V/Λ, a complex torus of dimension g, and X∨ = V ∨/Λ∨ its dual 2 torus. Let B Λ V ∨ be a real (constant) two form on X. Then B will decompose into parts ∈ B = B2,0 + B1,1 + B0,2 p,q p,q 0,2 2,0 1,1 1,1 0,2 2 0,1 where B Λ V ∨, B = B and B = B . Now B Λ V = 0,2 ∈ 2 0,1 0 2∈ ∼ H (X). Then it also represents a class Π Λ V ∼= H (X∨;Λ T1,0X). Let σ :Λ Λ U(1) denote the group 2-cocycle given∈ by ∧ → 2πıB(λ1,λ2) σ(λ1,λ2) = e . and form as above (Λ; σ), the twisted group algebra based on rapidly decreasing A 0, functions. Define the Dolbeault dga A •(Λ; σ) to be A (Λ; σ) Λ•V A ⊗ 1,0 where for λ (Λ; σ) we define ∈A ∂λ¯ =2πıλ D&(λ) (Λ; σ) V ⊗ ∈A ⊗ 1,0 0, We can then extend ∂¯ to the rest of •(Λ; σ) by the Leibniz rule. Let us reiterate the remarks above. Even though weA are defining the ∂¯ operator, we are using the

(1, 0) component of VC. This is because of duality. In the case of the trivial cocycle σ, this deﬁnition is meant to reconstruct the Dolbeault algebra on X∨. In this case, 0, 0,1 •(X∨) = •(X∨) Λ•T X∨. But A ∼ A ⊗ 0

0,1 ∨∨ T0 X∨ = V ∼= V1,0. To check the reasonableness of this deﬁnition we have

0, Proposition 5.16. If σ =1is the trivial cocycle, then the dga ( •(Λ; σ), ∂¯) 0, A is isomorphic to the Dolbeault dga ( •(X∨), ∂¯). A DUALITY AND EQUIVALENCE 27

0, We now show that the dga A =( •(Λ; σ), ∂¯, 0) is elliptic. Let A τ : (Λ; σ) C A → denote the continuous C-linear functional deﬁned by τ( aλλ)=a0. This is a trace, that is, τ(ab)=τ(ba) and in many cases it is the unique normalized trace on (Λ; σ). (It is unique when σ is “irrational” enough.), We note the following lemmaA whose proof is straightforward.

Lemma 5.17. For any ξ V ∨, the derivation ξ deﬁned by (5.8) has the property ∈ (5.9) τ ξ(f) =0 for all f (Λ; σ). ! " ∈A

Equip VC with a Hermitian inner product , : VC VC C. Let v1, . . . , vg and v1, . . . , vg be dual orthonormal bases of V and(· ·)V respectively.× → Equip V with C C∨ C a Hermitian structure. Then V1,0 and V0,1 inherit Hermitian structures as well. Let vi& and vi&& (i =1, . . . , g) be orthonormal bases of V1,0 and V0,1 respectively. We let D = (Λ; σ) ΛgV with A ⊗ 0,1 0,1 D: D D (Λ;σ) (Λ; σ) → ⊗A A defined by ¯ D(f v&&1,...,g )=∂(f) v&&1,...,g ⊗ { } ⊗ { } Recall that V1,0 is the anti-holomorphic cotangent space of the noncommutative complex torus and V0,1 is the holomorphic cotangent space. 0,k 0,g k Define ¯: (Λ; σ) D (Λ;σ) − by ∗ A → ⊗A A ¯(f vI& )=v&&1,...,g f ∗ vI& c ∗ ⊗ { } ⊗ ⊗ 0,g 2g Now note that D (Λ;σ) (Λ; σ) = (Λ; σ) and so we define ⊗A A ∼ A 0,g : D (Λ;σ) (Λ; σ) C ⊗A A → 4 Define 0 if I = 1, . . . , g (5.10) aλλ v& v&& = - { } ⊗ I ∧ I τ(a ) if I = 1, . . . , g 4 7 λ { } The following lemma is trivial to verify.

0,g Lemma 5.18. For all x D (Λ;σ) (Λ; σ) we have ∈ ⊗A A D(x)=0. 4 Theorem 5.19 (Serre duality for complex noncommutative tori). 0, ¯ (1) The dga A =( •(Λ; σ), ∂, 0 is elliptic with dualizing module (D, D), ¯, ). A ∗ (2) On the category 0, (Λ;σ) there is a Serre functor deﬁned by PA • 3 (E•, E) (E (Λ;σ) D, E # D) %→ ⊗A (3) In the case when σ =1, the Serre functor coincides with the usual Serre functor on X∨ using the isomorphism described in Proposition 5.16. 28 J. BLOCK

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Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA E-mail address: [email protected]