Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Let ${\mathcal E}$ be a holomorphic vector bundle on a complex manifold $X$ such that $\dim_{{\mathbb C}}X=n$. Given any continuous, basic Hochschild $2n$-cocycle $\psi_{2n}$ of the algebra ${\rm Diff}_n$ of formal holomorphic differential operators, one obtains a $2n$-form $f_{{\mathcal E},\psi_{2n}}(\mathcal D)$ from any holomorphic differential operator ${\mathcal D}$ on ${\mathcal E}$. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that $\int_X f_{{\mathcal E},\psi_{2n}}({\mathcal D})$ gives the Lefschetz number of $\mathcal D$ upto a constant independent of $X$ and ${\mathcal E}$. In addition, we obtain a"local"result generalizing the above statement. When $\psi_{2n}$ is the cocycle from [Duke Math. J. 127 (2005), 487-517], we obtain a new proof as well as a generalization of the Lefschetz number theorem of Engeli-Felder. We also obtain an analogous"local"result pertaining to B. Shoikhet's construction of the holomorphic noncommutative residue of a differential operator for trivial vector bundles on complex parallelizable manifolds. This enables us to give a rigorous construction of the holomorphic noncommutative residue of $\mathcal D$ defined by B. Shoikhet when ${\mathcal E}$ is an arbitrary vector bundle on an arbitrary compact complex manifold $X$. Our local result immediately yields a proof of a generalization of Conjecture 3.3 of [Geom. Funct. Anal. 11 (2001), 1096-1124].

1.1 Let X be a connected compact complex manifold such that dim C X = n. Let E be a holomorphic vector bundle on X. In this paper, the term "vector bundle" shall hence forth mean "holomorphic vector bundle". Let Diff(E) denote the sheaf of holomorphic differential operators on E. Let Ω 0,• X denote the Dolbeault resolution of the sheaf O X of holomorphic functions on X. Let Diff • (E) = Ω 0,• X ⊗ O X Diff(E) and let Diff • (E) = Γ(X, Diff • (E)). There is a suitable topology on Diff • (E) which we describe in Section 2. Recall that any global holomorphic differential operator D induces an endomorphism of the Dolbeault complex K • E := Γ(X, Ω • X ⊗ O X E) that commutes with the∂ differential. The Lefschetz number of D, also known as the supertrace of therefore makes sense. More generally, let E be a vector bundle with bounded geometry on an arbitrary complex manifold X of complex dimension n. The reader may refer to Section 2.2 for the definition of "bounded geometry". There is a complex hoch(Diff(E)) of "completed" Hochschild chains that one can associate with Diff • (E). This is a complex of (c-soft) sheaves of C-vector spaces on X. Let Γ c denote "sections with compact support". Suppose that α ∈ Γ c (X, hoch(Diff(E))) (see Section 2 for precise definitions). Let α 0 denote the component of α with antiholomorphic degree 0. For any t > 0, α 0 e −t∆ E is a trace class operator on the (graded) Hilbert space K • E L 2 of square integrable sections of K • E (Theorem 4, part 1). The supertrace str(α 0 e −t∆ E ) of α 0 e −t∆ E is given by the formula We may define the Lefschetz number of α to be lim t→∞ str(α 0 e −t∆ E ).
When X is compact and α = D is a global holomorphic differential operator on E, this coincides with the Lefschetz number of D defined earlier (any vector bundle on a compact manifold is of bounded geometry).
The subscript 'c' above indicates "compact supports". As a map of complexes of sheaves, f E,ψ 2n depends upon the choice of a Fedosov connection on B E . Nevertheless, as a map in the bounded derived category D b (Sh C [X]) of sheaves of C-vector spaces on X, f E,ψ 2n is independent of the choice of Fedosov connection on the bundle of fibrewise formal differential operators on E (see Proposition 15). Let τ 2n denote the normalized Hochschild cocycle from [10]. Let [..] denote "class in cohomology". Since HH 2n (Diff n ) is 1-dimensional, the ratio [ψ 2n ] [τ 2n ] makes sense.
1.3 One of the main results in this paper (proven in Section 3) is the following.
A special case of the above theorem, when α = D is a global holomorphic differential operator on X and when X is compact is as follows.

1.4
In particular, if we put ψ 2n to be the normalized Hochschild cocycle τ 2n of [10], the 2n-form f E,τ 2n (D) was denoted in [9] by χ 0 (D). We therefore obtain a different proof of the following Lefschetz number theorem of [9] as well as a generalization of this result.
We sketch in a remark Section 3.5 why the map α → X f E,ψ 2n (α) may be viewed as an "integral of ψ r 2n over X" in some sense. Upto this point, this paper completes our work in [17] and [18]. We feel that our proof of the Engeli-Felder formula (in [17], [18] and this paper) is "computationally lazy": our method indeed uses simpler analysis and less involved computations to prove this result. Moreover, our approach yields a new, more general result (Theorem 1) for free.
For any global holomorphic differential operator D on E, he refers to the number X Θ(E 11 (D)) as its holomorphic noncommutative residue (whenever X is compact). Since Ψ 2n+1 is not GL(n)basic, this cannot be done as directly as in [19] unless X is complex parallelizable and E is trivial (as per our understanding).
where C is a constant independent of X and E.
We prove Theorem 3 in Section 4.2. The methods used for this are similar to those used in [17] and [18]. In Section 4.2.5 and 4.2.6, we provide a rigorous (though not as explicit as we had originally hoped for) definition of B. Shoikhet's holomorphic noncommutative residue of a holomorphic differential operator D on E for the case when E is an arbitrary vector bundle over an arbitrary compact complex manifold X. Theorem 3 easily implies that this number is C times the Lefschetz number of E (C being the constant from Theorem 3). This settles conjecture 3.3 of [19] in greater generality.
Acknowledgements. I am grateful to Giovanni Felder and Thomas Willwacher for some very useful discussions. This work would not have reached its current form without their pointing out important shortcomings in earlier versions. I am also grateful to Boris Shoikhet for useful discussions. This was was done partly at Cornell University and partly at IHES. I am grateful to both these institutions for providing me with a congenial work atmosphere.
Convention 0. Throughout this paper, the term double complex will mean "complex of complexes". If {C p,q } is a double complex with horizontal and vertical differentials d h and d v respectively, the differential we use on the shifted total complex Tot(C)[k] will satisfy d( The Hochschild chain complex Γ(X, hoch(Diff(E))) and other preliminaries.
. Let A • be any DG-algebra. Let C • (A • ) denote the complex of Hochschild chains of A • . Whenever necessary, C • (A • ) will be converted to a cochain complex by inverting degrees. Note that the differential on C • (Diff • (E)(U )) extends to a differential on the graded vector space Let hoch(Diff(E)) denote the sheaf associated to the presheaf of complexes of C-vector spaces. Note that hoch(Diff(E)) is a complex whose terms are modules over the sheaf of smooth functions on X. It follows that H • (Γ(X, hoch(Diff(E)))) ≃ H • (X, hoch(Diff(E))). (2) Here, one must convert the chain complex hoch(Diff(E)) into a cochain complex by inverting degrees before taking hypercohomology.
Proof. Note that the differential on the Hochschild chain complex C • (Γ(U, Diff(E))) extends to a differential on the graded vector space Call the resulting complex C • (Γ(U, Diff(E))). One can then consider the sheaf C • (Diff(E)) associated to the presheaf of complexes of C-vector spaces. Note that the natural map of complexes from C • (Diff(E)) to hoch(Diff(E)) is a quasiisomorphism. This is because on any open subset U of X on which E and T 1,0 X are trivial, Γ(U k , Diff(E ⊠k ) is quasiisomorphic to Γ(U k , Diff • (E ⊠k ). The desired proposition then follows from (2) and the fact that C • (Diff(E)) is quasiisomorphic to the shifted constant sheaf C[2n]. This fact (see [17], Lemma 3) is also implicit in many earlier papers, for instance, [5].
Convention 1. Whenever we identify H i (Γ(X, hoch(Diff(E)))) with H 2n−i (X, C), we use the identification coming from a specific quasiisomorphism between C • (Diff(E)) and C[2n]. This quasiisomorphism is constructed as follows. "Morita invariance" yields a quasiisomorphism between C • (Diff(E)) and C • (Diff(X)) (see the proof of Lemma 3 of [17]). By a result of [5], C • (Diff(X)) is quasiisomorphic to C[2n]. The quasiisomorphism we shall use throughout this paper is the one taking the class of the normalized Hochschild 2n-cycle σ∈S 2n to 1[2n] on any open subset U of X with local holomorphic coordinates z 1 , .., z n . We remark that in the above formula, any permutation σ in S 2n permutes the last 2n factors leaving the first one fixed. We denote this identification by β E as we did in [17].
2.1.1 As in [17], we define a topology on Diff • (E) as follows. Let Diff ≤k,• (E) denote Γ(X, Ω 0,• X ⊗ Diff ≤k (E)) where Diff ≤k (E) denotes the sheaf of holomorphic differential operators on E of order ≤ k. Equip E and Ω 0,• X with Hermitian metrics. Equip Diff ≤k,• (E) with the topology generated by the family of seminorms {||.|| φ,K,s |K ⊂ X compact , s ∈ Γ(K, E ⊗ Ω • ), φ a C ∞ differential operator on E} given by The topology on Diff • (E) is the direct limit of the topologies on the Diff ≤k,• (E). More generally, for any open subset U of X, one can define a topology on Γ(U, Diff • (E)) in the same way.

2.2
Let E be a holomorphic vector bundle on an arbitrary connected complex manifold X. Let ∆ E denote the Laplacian of E (for the operator∂ E ). This depends on the choice of Hermitian metric for E as well as on a choice of a Hermitian metric for X. Recall that ∆ E = ∆ E + F where ∆ E is the Laplacian of a connection on E ( see Definition 2.4 of [2]) and F ∈ Γ(X, End(E)).
Definition. We say that E has bounded geometry if for some choice of Hermitian metric on E, there exists a connection ▽ E on E such that ∆ E = ∆ E + F , where ∆ E is the Laplacian of ▽ E and F ∈ Γ(X, End(E)), and all covariant derivatives of the curvature of ▽ E as well as those of F are bounded on X. In particular, any holomorphic vector bundle on a compact complex manifold has bounded geometry.
Let E be a vector bundle with bounded geometry on X. Let K • E L 2 denote the Hilbert space of square integrable sections of K • E . Then, e −t∆ E makes sense as an integral operator on K • E L 2 for any t > 0 (see [8]). Let Γ c denote the functor "sections with compact support". Suppose that α ∈ Γ c (X, hoch(Diff(E))) is a 0-chain. Let α 0 denote the component of α with antiholomorphic degree 0. We recall the main results of [17] and [18] in the following theorem.

Theorem 4.
(1) For any 0-chain α, α 0 e −t∆ E is a trace class operator on K • E L 2 for any t > 0.
where [α] denotes the class of α in H 2n c (X, C).

Remark 1.
Let us recall some aspects of [17] and [18]. When X is compact, the linear functional on Γ c (X, hoch(Diff(E))) is really an extension of the Feigin-Losev-Shoikhet Hochschild 0-cocycle of Diff • (E). We therefore, denote the linear functional given by (3) by I hoch FLS . To show that the Feigin-Losev-Shoikhet cocycle indeed gives a linear functional on H 0 (Γ(X, hoch(Diff(E)))) takes the sole computational effort in this whole program. The crux of this is a certain "estimate" of the Feigin-Losev-Shoikhet cocycle (Proposition 3 of [17]). This estimate is proven in "greater than usual" detail in [17]. It is done using nothing more than the simplest heat kernel estimates (Proposition 2.37 in [2]).
The second part of Theorem 4 can be exploited to extend the Feigin-Losev-Shoikhet cocycle to a linear functional on certain homology theories that are closely related to hoch(Diff(E)).
2.3 Cyclic homology... One obtains a "completed" version of the Bar complex of Γ(U, Diff • (E)) by replacing Γ(U, Diff • (E)) ⊗k by Γ(U k , Diff • (E ⊠k )). Sheafification then gives us a complex bar(Diff(E)) of sheaves on X having the same underlying graded vector space as hoch(Diff(E)). One can then construct Tsygan's double complex ...
The total complex of this double complex is denoted by Cycl(Diff(E)). Here, the operator t acts on Γ(U, Diff • (E)) ⊗k by where the D i are homogenous elements of Γ(U, Diff • (E)) of degree d i . t then extends to Γ(U k , Diff • (E ⊠k )) by continuity. It further extends to a map from bar(Diff(E)) to hoch(Diff(E)) by sheafification. Similarly, N is the extension of the endomorphism 1 + t + t 2 + ... + t k−1 of Γ(U, Diff • (E)) ⊗k to a endomorphism of the graded vector space hoch(Diff(E)). One can also consider the Connes complex Co(Diff(E)) := hoch(Diff(E))/(1 − t). There is a natural surjection Π : Cycl(Diff(E)) → Co(Diff(E)) given by the quotient map on the first column of the double complex (4) and 0 on other columns.
Proof. Fix an arbitrary integer n. Consider the sheaf C k associated to the presheaf Let hoch(Diff(E)) k n denote the sheaf of sections of the C k of degree n. As a complex, the n-th row in the double complex (4) is a direct sum of complexes C n • where every term of the complex C n k is the sheaf hoch(Diff(E)) k n . The differentials alternate between 1 − t and N . The proof of Theorem 2.1.5 in [14] shows that C n • is acyclic in degrees except 0. It follows that the n-th row of the double complex (4) is acyclic in all degrees except 0, where its homology is Co(Diff(E)) n . This proves the desired proposition.
Applying the linear functional I hoch FLS on the first column of the double complex of compactly supported global sections of the double complex (4), one obtains a linear functional I Cycl FLS on Γ c (X, Cycl(Diff(E))). Moreover, the endomorphism 1−t of the vector space Γ c (X, hoch(Diff(E)) 0 ) kills the direct summand Diff 0 (E). By definition, I hoch FLS (α) depends only on the component of α in Diff 0 (E). It follows that I hoch FLS vanishes on the image of 1 − t. It therefore, induces a linear functional on Γ c (X, Co(Diff(E))) which we will denote by I Co FLS . More explicitly, if α is a 0-chain in Γ c (X, Co(Diff(E))), I Co FLS (α) = lim t→∞ str(α 0 e −t∆ E ) where α 0 is the component of α in Diff 0 (E). The following proposition is a consequence of part 2 of Theorem 4.
Proof. We have already observed that I hoch FLS vanishes on the image of 1−t in Γ c (X, hoch(Diff(E)) 0 ). That it also vanishes on the image of the Hochschild boundary b of Γ c (X, hoch(Diff(E))) is implied by part 2 of Theorem 4. The desired proposition then follows from the construction of I Cycl FLS and I Co FLS .
The following proposition is immediate from the definition of I Co FLS .
Proposition 4. The following diagram commutes.

2.4
Lie algebra homology... We begin this subsection by reminding the reader of a convention that we will follow. Convention 2. Let g be any DG Lie algebra with differential of degree 1. By the term "complex of Lie chains of g", we shall refer to the total complex of the double complex {C p,q := (∧ p g) −q |p ≥ 1}. The horizontal differential is the Chevalley-Eilenberg differential and the vertical differential is that induced by the differential on g. Note that our coefficients are in the trivial g-module, and that we are brutally truncating the column of the usual double complex that lies in homological degree 0. In particular, if g is concentrated in degree 0, we do not consider Lie 0-chains of g.

Recall that Diff • (E) is a sheaf of topological algebras on X.
Over each open subset U of X, the topology on Γ(U, Diff • (E)) is given as in Section 2.1.1. It follows that gl r (Diff • (E)) is a sheaf of topological Lie algebras on X with the topology on Γ(U, gl r (Diff • (E))) induced by the topology on Γ(U, Diff • (E)) in the obvious way. Treating Γ(U, gl fin ∞ (Diff • (E))) as the direct limit of the topological vector spaces Γ(U, gl r (Diff • (E))) makes gl fin ∞ (Diff • (E)) a sheaf of topological Lie algebras on X. In this case, the complex of "completed" Lie chains C lie Here, σ acts on the factors gl fin ∞ (C) ⊗k and Γ(U k , Diff • (E ⊠k )) simultaneously. In an analogous fashion, one can construct the complex C lie ) since the colums of the corresponding double complexes are quasi-isomorphic.
Definition. We define Lie(Diff(E)) to be the complex of sheaves associated to the complex of presheaves In the proposition below, the subscript 'c' indicates "compact supports".
Proof. Recall from [11] that for any algebra A, we have an isomorphism ).
An analog of this result holds for the topological algebra Γ(U, Diff(E)) (see Section 2.3.3 of [4] for the corresponding assertion in the C ∞ case). Recall from [22], [17] that if E is trivial on U and if U has holomorphic coordinates, then the completed cyclic homology of Γ(U, The desired proposition now follows once note that C lie is a direct sum of copies of the constant sheaf C concentrated in homological degrees ≥ 2n + 1. By Proposition 3.2.2 of [13],

2.4.2
For any algebra A, we have a map of complexes where the right hand side is the Connes complex of A (the quotient of the Hochschild chain complex of A by the image of 1 − t). To be explicit, a Lie k for N sufficiently large. One then applies the generalized trace map to obtain a k chain in C λ It is easily verified that the map L extends by continuity to a map of complexes from C lie • (Γ(U, gl fin ∞ (Diff • (E)))) [1] to the completed Connes complex for Γ(U, Diff • (E)). By sheafification followed by taking global sections, it further yields a map of complexes from Γ c (X, Lie(Diff(E)) [1]) to Γ c (X, Co(Diff(E))) which we shall continue to denote by L.
Proof. Recall from [14] that for any algebra A, L induces an isomorphism between the primitive part of H lie • (gl fin ∞ (A)) and HC •−1 (A). For the topological algebra Γ(U, Diff • (E)) on a sufficiently small open subset U of X, it induces an isomorphism between the primitive part of ))) and the completed cyclic homology of Γ(U, Diff • (E)). In particular, it induces an isomorphism between H 2n ( C lie ))) both of which are isomorphic to C. The desired proposition follows from this.
It follows from Propositions 4 and 6 that the map I Co FLS • L induces a nonzero linear functional on H 0 (Γ c (X, Lie(Diff(E))[1])). We will denote this linear functional by I lie FLS at the level of chains as well as on homology. Note that if α is a 0-chain in Γ c (X, Lie(Diff(E)) [1]), the component of and Tr is the usual matrix trace. Therefore, Let F be another holomorphic vector bundle on X. There is a natural map ι : Proposition 7. The following diagram commutes.
). Clearly, Tr(ῑ(α) 0 ) = ι(Tr(α 0 )). It then follows immediately that provided the Hermitian metric on E ⊕ F is chosen by retaining that on E and choosing an arbitrary Hermitian metric on F. Part 2 of Theorem 4 however, implies that on homology, I lie,E⊕F FLS is independent of the choice of Hermitian metric on E ⊕ F. This proves the desired proposition.
3 Differential forms computing the Lefschetz number.
3.1 Fedosov differentials. The material in this subsection is a by and large a rehash of material from [9] and [19]. We extend a Gelfand-Fuks type construction from [9] in this section. This construction is a version of a construction that was originally done in [12] by I. M. Gelfand and D. B. Fuks for Lie cocycles with trivial coefficients for the Lie algebra of smooth vector fields on a smooth manifold. We begin by recalling the construction from [9].
3.1.1 Let r be the rank of E. Let J p E denote the bundle of p-jets of local trivializations of E. In particular, J 1 E is the extended frame bundle whose fiber over each x ∈ X consists of the set of all pairs comprising a basis of T 1,0 X,x and a basis of E x . The group G := GL(n, C) × GL(r, C) acts on the right on J p E for each p. More specifically, given a local isomorphism of bundles C n × C r → E, GL(n, C) acts by linear coordinate transformations on C n and GL(r, C) acts by linear transformations on C r . This makes Since the fibres of this map are contractible, there exists a smooth section ϕ of J ∞ E/G over X. This is equivalent to a G-equivariant section of J ∞ E over J 1 E. This section is unique upto smooth homotopy.
. This semidirect product comes from the action of W n on gl(O n ) by derivations. We recall from [9] that J ∞ E is a principal W n,r bundle. To be precise, there is a map of Lie algebras from W n,r to the Lie algebra of holomorphic vector fields on J ∞ E which yields an isomorphism W n,r → T 1,0 φ J ∞ E for each φ ∈ J ∞ E. This is equivalent to a W n,r -valued holomorphic 1-form Ω on J ∞ E satisfying the Maurer-Cartan equation Since there is a natural map of Lie algebras W n,r → M r (Diff n ), Ω may be viewed as a Maurer-Cartan form with values in M r (Diff n ).

3.1.3
Consider the bundle B E := J 1 E × G M r (Diff n ) of algebras with fiberwise product. Any trivialization of J 1 E over an open subset U of X yields an isomorphism of algebras More generally, a trivialization of J 1 E over U yields an isomorphism of graded algebras The G equivariant section ϕ : J 1 E → J ∞ E allows us to pull back Ω. This gives us a one form ω := ϕ * Ω ∈ Ω 1 (J 1 E, M r (Diff n )) G . This in turn yields a flat connection D on the bundle of algebras B E . Let us be explicit here. Given a trivialization of J 1 E over U , ω descends to a M r (Diff n ) valued one form ω U on U satisfying the Maurer-Cartan equation. The isomorphism (6) identified the connection D with the "twisted" De-Rham differential d+ [ω U , −]. More generally, Over any open subset U of X, the degree 0 sectionsD of Ω • (B E ) satisfying are in one-to-one correspondence with holomorphic differential operators on E over U . The differential D is a Fedosov differential on the sheaf Ω • (B E ) of DG-algebras on X.

3.1.4
There is a different construction of a Fedosov differential D on Ω • (B E ) due to Calaque and Rossi [6] (see also Dolgushev,[7]) in the case when E = O X . This goes through with minor modifications for the case when E is arbitrary. This is a more "careful" construction of a Fedosov differential: it has the following special property (Part 2 of Theorem 10.5 of [6]: note that we do not need Part 1 of this result).

Theorem 5.
There is a map of sheaves of DG-algebras Given that the construction of the Fedosov differential D from [6] is a priori different from that of [9], one would like to know whether the Fedosov differential D from [6] can also be obtained via the construction from [9] outlined earlier. The reason why we need this relationship will become clear in later sections. The following paragraphs give a partial affirmative answer to this question that suffices for us. To be precise, in the next paragraph shows that there exists a G-equivariant section ϕ : J 1 E → J ∞ E associated with any Fedosov differential on Ω • (B E ). Moreover, since the above isomorphism preserves the order of the differential operator, we obtain an isomorphism θ : This corresponds to an element in the fibre of J ∞ E over x. We therefore, obtain a section ϕ : It is easy to verify that this section is G-equivariant. Note that the isomorphism θ induces an isomorphismθ from Jets x Diff(E) to M r (Diff n ).
Proof. Note that M r (O n ) and Id ⊗ W n generate M r (Diff n ) as an algebra. It therefore suffices to check thatθ −1 and Θ −1 coincide on M r (O n ) and Id ⊗ W n .Observe that the image of W n,r := W n ⋉ gl r (O n ) in M r (Diff n ) is precisely the space of all elements of the form X = D + f for some D ∈ Id ⊗ W n and some f ∈ M r (O n ). We shall continue to denote this image by W n,r .
We first prove the proposition assuming our claim. Indeed, Θ −1 restricted to M r (O n ) coincides withθ −1 restricted to M r (O n ) (both of which coincide with θ −1 ). For any X ∈ W n,r and for It follows that for any X ∈ W n,r ,θ −1 (X) − Θ −1 (X) commutes with all elements of Jets x End(E).
In other words,θ −1 (X) − Θ −1 (X) must be an element of θ −1 (id ⊗ O n ) for all X ∈ W n,r . In particular, if X ∈ id ⊗ W n , thenθ −1 (X) = Θ −1 (X) + θ −1 (g) for some g ∈ id ⊗ O n . In this case, It follows thatθ −1 ( This shows thatθ −1 and Θ −1 coincide on id ⊗ W n as well. Our claim remains to be proven. Recall (from [14] for instance) that for any algebra A, In our case, the composite map is an isomorphism. The first arrow above is the Hochschild-Kostant-Rosenberg map. The second arrow above is the cotrace. It follows that for any derivation φ of M r (Diff n ), there is an element v of W n such that the image of v under the above composite map coincides with that of φ in 3.2 A Gelfand-Fuks type construction. We now recall and extend a construction from [9].
as graded vector spaces. The differential on C • (A) extends naturally to a differential on C Π making the natural map of graded vector spaces a map of complexes. One has the following proposition due to Engeli and Felder [9]. Though [9] used normalized Hochschild chains, their proof goes through in the current context as well.
is a term by term isomorphism of complexes.

3.2.2
Recall that GL(n, C) acts on a formal neighborhood of the origin by linear coordinate changes. This induces an action of GL(n, C) on Diff n . The derivative of this action embeds the Lie algebra gl n in Diff n as the Lie subalgebra of operators of the form j,k a j,k y k ∂ j . We say that the normalized cocycle α ∈ C p (Diff n ) is GL(n, C)-basic if α is GL(n, C)-invariant and if p i=1 (−1) i+1 α(a 0 , ..., a i−1 , a, a i+1 , .., a p ) = 0 for any a ∈ gl n and any a 0 , ..., a p ∈ Diff n . More generally, G := GL(n, C) × GL(r, C) acts on M r (Diff n ). The action of GL(n, C) on M r (Diff n ) is induced by the action of GL(n, C) on Diff n and GL(r, C) acts on M r (Diff n ) by conjugation. The derivative of this action embeds g := gl n ⊕ gl r as a Lie subalgebra of M r (Diff n ) of elements of the form j,k a j,k z k ∂ j ⊗ id r×r + B where B ∈ M r (C). Again, a cocycle α in C p (M r (Diff n )) is said to be G basic if α is G-invariant and p i=1 (−1) i+1 α(a 0 , ..., a i−1 , a, a i+1 , .., a p ) = 0 for any a ∈ g and any a 0 , ..., a p ∈ M r (Diff n ). The following proposition is immediate from equation (1).
We also recall that a cocycle α ∈ C p (M r (Diff n )) is said to be continuous if it depends only on finitely many Taylor coefficients of its arguments.
Proposition 11. The map (7) is a map of complexes (with Ω 2n−p (U ) equipped with the differential (−1) p d DR ).
Observe that there is a natural map of complexes C Explicitly, the map (8) maps a chain µ ∈ C p (Ω • (U, B E ), D) to whereμ is the image of µ in C Π p (Ω • ω (U, M r (Diff n ))). (8) is independent of the trivialization of J 1 E used.

Proposition 12. The map
Proof. A different trivialization of J 1 E over U differs from the chosen one by a gauge change g : U → G. A section µ of Ω • (U, B E ) transforms into g.µ. The Maurer-Cartan form ω is replaced by g.ω − dg.g −1 . That ψ r 2n (α × k ±(ω) k ) = ψ r 2n (g.α × k ±(g.ω − dg.g −1 ) k ) is immediate from the fact that ψ r 2n is G-basic. This proves the desired proposition.
It follows that the map (8) gives a map of complexes of presheaves Note that whatever we said so far is true for any Fedosov differential no matter how it was constructed. Of course, the map of complexes of presheaves (9) depends on the Fedosov differential D. It follows from Theorem 5 that we obtain a composite map of complexes of presheaves has the structure of a (graded) locally convex topological vector space. One can verify without difficulty that the subcomplex C • (Diff • (E)(U )) is dense in C • (Diff • (E)(U )) and that the differential of C • (Diff • (E)(U )) is continuous. Similarly, Ω 2n−• (U ) has the structure of a (graded) complete locally convex topological vector space and the de-Rham differential is continuous with respect to this topology.
Suppose that f E is continuous. Then, f E extends to a map of complexes from C • (Diff • (E)(U )) to Ω 2n−• (U ) for each U . This can be easily see to yield a map of complexes of presheaves On sheafification, f yields a map of complexes of sheaves hoch(Diff(E)) → Ω 2n−• X proving the desired proposition. Indeed, continuity of f E follows from continuity of ψ r 2n . Since ψ r 2n is continuous, C l -norms of f (a 0 ⊗ ... ⊗ a k ) over a compact subsets of U are estimated by C l ′norms of finitely many Taylor coefficients of the imagesâ i of the a i in Ω • (U, B E ) over compact subsets of U .
The map of complexes of sheaves obtained in Proposition 13 depends on the cocycle ψ 2n . We shall denote this map by f E,ψ 2n . Recall that there is a natural map of complexes of sheaves C • (Diff(E)) → hoch(Diff(E)). Let U be a subset of X with local holomorphic coordinates z 1 , .., z n such that E is trivial over U . By Convention 1, the standard generator for the homology of Γ(U, C • (Diff(E))) is a cycle c E (U ) mapping to the normalized Hochschild 2n-cycle Let c r 2n denote the normalized Hochschild 2n-cycle σ∈S 2n sgn(σ)σ(Id r×r ⊗ 1, Id r×r ⊗ y 1 , Id r×r ⊗ ∂ ∂y 1 , ....., Id r×r ⊗ y n , Id r×r ⊗ ∂ ∂y n ) of M r (Diff n ). Note that ψ r 2n (c r 2n ) makes sense since ψ r 2n is G-basic.
Proposition 14. f E,ψ 2n (c E (U )) = ψ r 2n (c r 2n ). Proof. Note that any element D of Diff(E)(U ) gives a holomorphic M r (Diff n )-valued functionD on J ∞ E| U , satisfying dD + [Ω,D] = 0 where Ω is the Maurer-Cartan form from Section 3.1.2. It follows that there is a map of complexes The last arrow is by Proposition 9. Evaluation at ψ r 2n followed by applying the involution multiplying p-forms by (−1) ⌊ p 2 ⌋ therefore yields a map of complexes Given a section ϕ : J 1 E → J ∞ E and a trivialization of J 1 E over U , one has a composite map The middle arrow above is pullback by the section ϕ : J 1 E → J ∞ E. The rightmost arrow above is pullback by the section of J 1 E arising out of the trivialization of J 1 E that we have chosen over U . Any section ϕ : J 1 E| U → J ∞ E| U is unique upto homotopy. It follows from this that the composite map (11) is unique upto homotopy for a fixed trivialization of J 1 E over U . This fact and the proof of Proposition 12 together imply that the image of c E (U ) under (11) is independent of the precise choice of ϕ and of trivialization of J 1 E over U . To compute it, we could choose ϕ to be the section taking ∂ ∂z i to the formal derivative ∂ ∂y i and z i to y i + a i at (a 1 , ...., a n ). In this situation, the image of c E (U ) is indeed seen to be ψ r 2n (c r 2n ) in homological degree 2n.

Properties of
In particular, the map f E in D b (Sh C [X]) is independent of the choice of Fedosov connection on B E .
Proof. As objects in D b (Sh C [X]), hoch(Diff(E)) as well as Ω 2n−• X are isomorphic to the shifted constant sheaf C[2n]. Since C is an injective object in the category of sheaves of C-vector spaces on X (since it is flabby), It follows that it is enough to verify this proposition for each open subset U of X with local holomorphic coordinates on which E is trivial. For such a U , the generalized trace map maps c E (U ) to a 2n-cycle mapping to the normalized Hochschild 2n-cycle It follows from Convention 1, Section 2.1 that β E (c E (U )) = r. Therefore, 1 (1). Further, since τ 2n (c 2n ) = 1 (see [10]), τ r 2n (c r 2n ) = r by (1). Therefore, 1 ] . This proves the desired proposition.
Since f E,ψ 2n is a map of complexes of sheaves, the map descends to a (nonzero) linear functional on H 0 (Γ c (X, hoch(Diff(E)))). We shall denote this linear functional by X f E,ψ 2n .

Proof of Theorem 1. Proposition 15 implies that
in H 2n c (X, C) for any 0-cycle α in Γ c (X, hoch(Diff(E)). Theorem 1 follows immediately from this observation and Theorem 4.

Remarks. f E,ψ 2n induces a map of complexes
This can equivalently be viewed as a family Θ i ∈ Ω i (C 2n−i (Diff • (E))) of cochain valued forms satisfying the differential equations where δ is the differential on the Hochschild cochain complex C 2n−i (Diff • (E)). In particular, Θ 2n is a 2n-form with values in C 0 (Diff • (E)). When X is compact, X Θ 2n is precisely the Hochschild 0-cocycle (12). This viewpoint seeing (12) as coming from "integrating ψ r 2n over X" is taken by [19].
More generally, there is a modified cyclic chain complex Cycl(Diff(E)) related closely to hoch(Diff(E)) (see Section 2.3). The construction of this section can be repeated for a continuous G-basic cyclic 2n + 2p-cocycle ν 2n+2p (see [16], [21]). One obtains a map of complexes as a result. When X is compact, and k ≥ p, the above map on the 2k-th homology yields a map It would be interesting to understand this map further.
We begin by outlining the construction of Ψ 2n+1 in Section 4.1. Section 4.2 is devoted to proving Theorem 3. We therefore call it a trace and denote it by Tr ΨDiffn : ΨDiff n → C (see [1]). Recall that the "usual" matrix trace Tr gl fin ∞ yields a linear map from gl fin ∞ (A) to A for any algebra A. It can then be verified that Tr ΨDiffn • Tr gl fin ∞ yields a trace on the algebra gl fin ∞ (ΨDiff n ).
Consider the set S of all markings of the interval [1, 2n − 1] such that (i) Only finitely many integral points are marked. (ii) The distance between any two distinct marked points is at least 2. Note that the "empty" marking where no point is marked is also an element of S.
Let t ∈ S be a marking of [1, 2n − 1] marking the integers i 1 , ..., i k . Define where if j is marked, P j,t = A j Q j,j+1 and P j+1,t = A j+1 P j,t = D j (A j ) if j and j − 1 are not marked and j = 2n + 1 Note that 0 and 2n should be thought of as unmarked by default in the above formula. If t is the "empty" marking, is a 2n + 1-cocycle in C 2n+1 Lie (gl fin ∞ (ΨDiff n ); C).

Note that for any algebra A, we have isomorphisms
of algebras. Taking the direct limit of these isomorphisms, we obtain a map of algebras (and therefore, Lie algebras) It follows that Ψ 2n+1 yields a cocycle in C 2n+1 lie (gl fin ∞ (M m (Diff n )), C) as well. We denote this cocycle by Ψ m 2n+1 . For Proof. It is enough to show that for any t ∈ S, The summands on the left hand side of (4.1) are of the form Tr ΨDiffn • Tr gl fin It is easy to see that this summand does not change if each X i of the form X i = (A j ⊕ B j ) ⊗ id is replaced by A j ⊗ id. Doing this however transforms the sum on the left hand side to that on the right hand side.

4.2.1
Let g be a DG-Lie algebra. Then, {C p,q := (∧ p g) −q |p ≥ 1} becomes a double complex whose horizontal differential is the Chevalley-Eilenberg differential and whose vertical differential is that induced by the differential intrinsic to g. We denote the complex Tot ⊕ (C •,• ) by C lie • (g). Similarly we denote Tot Π (C •,• ) by C Π,lie • (g). There is a natural map of complexes from C lie • (g) to C Π,lie • (g). Let ω ∈ g 1 be a Maurer-Cartan element. Let g ω denote the twisted Lie algebra whose underlying differential is d + [ω, −].
Proof. Since this proposition is completely analogous to Proposition 2.4 of [9], we shall only sketch the proof. Denote the differential of g by d. Let d CE denote the Chevalley-Eilenberg differential. Let (ω) j := ω ∧ .... ∧ ω j times .
Step 1. One first notes that The middle equality above is because ω is a Maurer-Cartan element. It follows that if φ j : Step 2. Let g 0 , ..., g k be homogenous elements of g. Let d i denote the degree of g i . Let G := g 0 ∧ ... ∧ g k . One then verifies (by a direct calculation) that The desired proposition follows from equations (14) and (15) after inserting the relevant definitions and summing over k.
Notation. In some situations in this section, we find it better to specify the differential of a DG-Lie algebra: if d is the differential on a DG-Lie algebra g we often denote C lie • (g) and C Π,lie . Also, ω ⊗ 1 N is a Maurer-Cartan element in gl fin ∞ (Ω • (U, M r (Diff n ))) for sufficiently large N . We will continue to denote this element by ω for notational brevity.
One therefore has the following composite map of complexes.
The horizontal arrow on top is from the isomorphism of (Ω • (U, B E ), D) with (Ω • (U, M r (Diff n )), d+ [ω, −]) induced by ϕ. The vertical arrow on the right is the natural map mentioned in Section 4.2.1. The horizontal arrow on the bottom is from Proposition 17. Let Ξ 2n+1 be any continuous 2n + 1 cocycle in C 2n+1 lie (gl fin ∞ (M r (Diff n )), C). As in Proposition 11, Section 3.2.3, evaluation at Ξ 2n+1 yields a map of complexes from C Π,lie [1] to Ω 2n−• (U ) (with the differential on Ω 2n−p being (−1) p d DR ). Composing this map with θ, and applying the involution multiplying a p-form by (−1) ⌊ p 2 ⌋ , we obtain a map of complexes Explicitly, if µ ∈ C lie p (gl fin ∞ (Ω • (U, B E ), D)), whereμ is the image of µ in C lie p (gl fin ∞ (Ω • (U, M r (Diff n )), d + [ω, −])). In In particular, we obtain a map of complexes Proof. This follows from the continuity of Ξ 2n+1 . The argument proving this is completely analogous to the proof of Proposition 13. However, since Ξ 2n+1 may not be G := GL(n, C) × GL(r, C)-basic, we can only guarantee the existence of a map of complexes of sheaves over U .
Proof. The obstruction to globalizing λ(Ξ 2n+1 ) comes from the fact that there is no consistent way of choosing a section of J 1 E over X in general.
Proof. From the discussion in Section 2.4.1, Lie(Diff(E)) [1] is isomorphic to ). Since C is an injective object in the category of sheaves of C-vector spaces, Hom D b (Sh C [U ]) (V, C[2n]) = 0. It therefore, suffices to show that λ(Ξ 2n+1 ) applied to a fixed 2n cycle generating the 2n-st homology of Γ(U, Lie(Diff(E)) [1]) is independent of the choice of Fedosov differential.
As in the proof of Proposition 14, given any (G-equivariant) section ϕ : J 1 E → J ∞ E, one has a composite map of complexes The "restriction" of the first map above to C lie • (Γ(U, gl fin ∞ (Diff(E)))) [1] is constructed just like the analogous map in the proof of Proposition 14. The middle arrow is ϕ * where ϕ : Recall that there is a natural map of complexes from C lie • (Γ(U, gl fin ∞ (Diff(E))))[1] to Γ(U, Lie(Diff(E)) [1]) (and similarly for E ⊕ F). As observed while proving Proposition 5, this map is an isomorphism on homology, and the constant sheaf of U corresponding to the homology of C lie • (Γ(U, Diff(E))) [1] is isomorphic to Lie(Diff(E)) [1] in D b (Sh C [U ]). It therefore, suffices to show that the following diagram commutes upto homology.  (Diff(U )). The (G-equivariant) section of J ∞ E we choose is the one that maps ∂ ∂z i to ∂ ∂y i and f (z 1 , ..., z n ) to (a 1 , .., a n ) → I ∂f ∂z I | (a 1 ,..,an) y I .
Let D ∈ Γ(U, gl fin ∞ (Diff(E))) be arbitrary. LetD denote the flat section of C ∞ (U, gl fin ∞ (M r (Diff n ))) corresponding to D. Note that by our choices of sections of J ∞ E and J ∞ (E ⊕ F), ι(D) = ι r,r+sD .
It follows from equation (18) and Proposition 16 that the following diagram commutes literally (not just upto homology).
Since C lie • (Γ(U, gl fin ∞ (Diff(E)))) [1] is dense in C lie • (Γ(U, gl fin ∞ (Diff(E)))) [1] (and similarly for E ⊕F), and since all the maps involved in diagram (21) are continuous, commutativity of diagram (20) follows (note that this "commutativity on the nose" of diagram (20) is with our convenient choice of section of J ∞ E| U over J 1 E| U as well as our choice trivialization of J 1 E over U ). This proves the desired proposition.
We remark that our proof of Propositions 20,21 and 22 go through for arbitrary complex parallelizable manifolds.

Proof of Theorem 3.
For this subsection, we shall assume that X is complex parallelizable and E has bounded geometry. By propositions 20 and 21, for any choice used in the construction of λ(Ψ r 2n+1 ), the map induces the same map on the 0-th homology of Γ c (X, Lie(Diff(E)) [1]). On the other hand, we saw in Section 2.4.2 that I lie,E FLS induces a map on the 0-th homology of Γ c (X, Lie(Diff(E))[1]) as well. By Corollary 1, H 0 (Γ c (X, Lie(Diff(E))[1])) is a 1-dimensional C-vector space. It follows that as linear functionals on H 0 (Γ c (X, Lie(Diff(E))[1])), X λ(Ψ r 2n+1 ) = C(X, E).I lie,E FLS .
Propositions 7 and 22 together with the nontriviality of I lie,E FLS on homology imply that C(X, E) = C(X, E ⊕ F) for any vector bundle F with bounded geometry on X. This shows that C(X, E) is independent of E.
For the rest of this subsection assume that E = O X . Let U be an open subset of X with holomorphic coordinates that identify U with an open disk in C n . Choose a nontrivial 0-cycle α of Γ c (U, Lie(Diff(E)) [1]). Note that after making the necessary choices, the construction of λ(Ψ 2n+1 ) is local in nature. It follows that U λ(Ψ 2n+1 )(α) = X λ(Ψ 2n+1 )(j * α) (22) where j : U → X is the natural inclusion.
Further, if φ is an element of Diff 0 (X) that is compactly supported on U , str(φe −t∆ X ) = str(φe −t∆ U ) for any t > 0. In the right hand side of the above equation, we think of φ and e −t∆ U as endomorphisms of the space of square integrable sections of K • O U . To see this, note that if p t (x, y) is the kernel of e −t∆ X , str(φe −t∆ X ) = X str(φp t (x, x))|dx| = U str(φp t (x, x))|dx| = str(φe −t∆ U ).
The second equality above is because φ is compactly supported on U . The third equality above is because the heat kernel on U is unique once the choice of Hermitian metric on K • O U is fixed (see [8]). It follows from equation (22) (in the case when E = O X ) and equation (23) that C(X, E) is independent of X as well. This proves Theorem 3.

4.2.5
Once again, let X be an arbitrary compact complex manifold. Let F • be a complex of sheaves on X such that each F i is a module over the sheaf of smooth functions on X. Suppose also that F • is in D b (Sh C [X]). Let U := {U i } be a finite good cover of X. Consider the double complexČ(U, F • ) whereČ p,q (U, F • ) = ⊕ j 1 <...<jp Γ c (U j 1 ∩ ... ∩ U jp , F q ).
Here, (α) i 1 ,..,i k denotes the component of a chain α ∈Č k,l (U, F • ) in Γ c (U i 1 ∩...∩U i k , F l ). We also follow the convention that if i 1 < ... < i k , (α) i σ(1) ,..,i σ(k) = sgn(σ)(α) i 1 ,..,i k for any permutation σ of 1, .., k. The following proposition is a modification of Proposition 12.12 of Bott and Tu [3]. The subscript 'c' on the right hand side in this proposition stands for "compact support". The chain complex F • needs to be converted into a cochain complex by inverting degrees in order to make sense of the hypercohomology in the above proposition.
Proof. Since each F i is a module over the sheaf of smooth functions on X, H −p c (X, F • ) ≃ H p (Γ c (X, F • )). The second part of this proposition is therefore immediate from the first. The first part of this proposition follows from the fact that the q-th row of the double complex C(U, F • ) is a resolution of Γ c (X, F q ). This can be proven by following the proof of Proposition 12.12 in [3].
Proposition 24. Suppose that F • is acyclic in negative degrees. If X is compact, there exist 0-cycles α i of Γ c (U i , F • ) such that i [α i ] = [α] in H 0 (Γ c (X, F • )).

4.2.6
Shoikhet's holomorphic noncommutative residue: definition. Let X be an arbitrary compact complex manifold and let E be an arbitrary vector bundle on X. Let U := {U i } be a finite good cover of X. Let D be a global holomorphic differential operator on E. Note that E 11 (D) is a 0-cycle in Γ c (X, Lie(Diff(E)) [1]). Applying proposition 24 to the complex Lie(Diff(E)) [1] of sheaves on X, we see that there exist 0-cycles α i in Γ c (U i , Lie(Diff(E)) [1]) such that i [α i ] = [E 11 (D)] in H 0 (Γ c (X, Lie(Diff(E))[1])). Define the holomorphic noncommutative residue of D to be the sum One can make the choices used to define λ(Ψ r 2n+1 ) in the U i arbitrarily. We also remark that λ(Ψ r 2n+1 ) is not globally defined on X in a direct way (at least as far as we can see). By This proves conjecture 3.3 of [19] in greater generality.