Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

Let E be a holomorphic vector bundle on a complex manifold X such that dimCX=n. Given any continuous, basic Hochschild 2n-cocycle ψ2n of the algebra Diffn of formal holomorphic differential operators, one obtains a 2n-form fε,ψ2n(D) from any holomorphic differential operator D on E. We apply our ear...

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Datum:	2011
1. Verfasser:	Ramadoss, A.C.
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spelling	irk-123456789-1467752019-02-12T01:23:38Z Integration of Cocycles and Lefschetz Number Formulae for Differential Operators Ramadoss, A.C. Let E be a holomorphic vector bundle on a complex manifold X such that dimCX=n. Given any continuous, basic Hochschild 2n-cocycle ψ2n of the algebra Diffn of formal holomorphic differential operators, one obtains a 2n-form fε,ψ2n(D) from any holomorphic differential operator D on E. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that ∫X fε,ψ2n(D) gives the Lefschetz number of D upto a constant independent of X and ε. In addition, we obtain a ''local'' result generalizing the above statement. When ψ2n 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 D defined by B. Shoikhet when 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]. 2011 Article Integration of Cocycles and Lefschetz Number Formulae for Differential Operators / A.C. Ramadoss // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16E40; 32L05; 32C38; 58J42 DOI:10.3842/SIGMA.2011.010 http://dspace.nbuv.gov.ua/handle/123456789/146775 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description	Let E be a holomorphic vector bundle on a complex manifold X such that dimCX=n. Given any continuous, basic Hochschild 2n-cocycle ψ2n of the algebra Diffn of formal holomorphic differential operators, one obtains a 2n-form fε,ψ2n(D) from any holomorphic differential operator D on E. We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405-448; J. Noncommut. Geom. 3 (2009), 27-45] to show that ∫X fε,ψ2n(D) gives the Lefschetz number of D upto a constant independent of X and ε. In addition, we obtain a ''local'' result generalizing the above statement. When ψ2n 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 D defined by B. Shoikhet when 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].
format	Article
author	Ramadoss, A.C.
spellingShingle	Ramadoss, A.C. Integration of Cocycles and Lefschetz Number Formulae for Differential Operators Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ramadoss, A.C.
author_sort	Ramadoss, A.C.
title	Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
title_short	Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
title_full	Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
title_fullStr	Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
title_full_unstemmed	Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
title_sort	integration of cocycles and lefschetz number formulae for differential operators
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146775
citation_txt	Integration of Cocycles and Lefschetz Number Formulae for Differential Operators / A.C. Ramadoss // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 23 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT ramadossac integrationofcocyclesandlefschetznumberformulaefordifferentialoperators
first_indexed	2025-07-11T00:33:34Z
last_indexed	2025-07-11T00:33:34Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 010, 26 pages Integration of Cocycles and Lefschetz Number Formulae for Differential Operators Ajay C. RAMADOSS Department Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland E-mail: ajay.ramadoss@math.ethz.ch Received August 12, 2010, in final form January 07, 2011; Published online January 18, 2011 doi:10.3842/SIGMA.2011.010 Abstract. Let E be a holomorphic vector bundle on a complex manifold X such that dimCX = n. Given any continuous, basic Hochschild 2n-cocycle ψ2n of the algebra Diffn of formal holomorphic differential operators, one obtains a 2n-form fE,ψ2n(D) from any holo- morphic differential operator D on E . We apply our earlier results [J. Noncommut. Geom. 2 (2008), 405–448; J. Noncommut. Geom. 3 (2009), 27–45] to show that ∫ X fE,ψ2n (D) gives the Lefschetz number of D upto a constant independent of X and E . In addition, we obtain a “local” result generalizing the above statement. When ψ2n 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 D defined by B. Shoikhet when 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]. Key words: Hochschild homology; Lie algebra homology; Lefschetz number; Fedosov con- nection; trace density; holomorphic noncommutative residue 2010 Mathematics Subject Classification: 16E40; 32L05; 32C38; 58J42 1 Introduction 1.1. Let X be a connected compact complex manifold such that dimCX = n. Let E be a holo- morphic 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 Ω0,• X denote the Dolbeault resolution of the sheaf OX of holomorphic functions on X. Let Diff•(E) = Ω0,• X ⊗OX 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,Ω0,• X ⊗OX E) that commutes with the ∂̄ differential. The Lefschetz number of D, also known as the supertrace of D, str(D) := ∑ i (−1)iTr(D)\|Hi(X,E) 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 Γc denote “sections with compact support”. Suppose that α ∈ Γc(X, ˜hoch(Diff(E))) (see Section 2 for precise definitions). Let α0 denote the component of α mailto:ajay.ramadoss@math.ethz.ch http://dx.doi.org/10.3842/SIGMA.2011.010 2 A.C. Ramadoss with antiholomorphic degree 0. For any t > 0, α0e−t∆E is a trace class operator on the (graded) Hilbert space K•E L2 of square integrable elements of K•E (Theorem 4, part 1). The supertrace str(α0e −t∆E ) of α0e−t∆E is given by the formula str ( α0e−t∆E ) = ∑ i (−1)iTr ( α0e−t∆E ) \|KiE,L2 . We may define the Lefschetz number of α to be lim t→∞ str ( α0e −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). 1.2. Let Diffn denote the algebra C[[y1, . . . , yn]][∂1, . . . , ∂n] (∂i := ∂ ∂yi ) of formal holomorphic differential operators. Denote the complex of continuous normalized Hochschild cochains of Diffn with coefficients in Diff∗n by C•(Diffn). Let ψ2n be a cocycle in C2n(Diffn). For an al- gebra A, let Mr(A) denote the algebra of r× r matrices with entries in A. Then ψ2n extends to a cocycle ψr2n in C2n(Mr(Diffn)) by the formula ψr2n(A0 ⊗D0, . . . , A2n+1 ⊗D2n+1) = tr(A0 · · ·A2n+1)ψ2n(D0, . . . , D2n+1). (1) Note that the group G := GL(n,C) × GL(r,C) acts on the algebra Mr(Diffn) with GL(r,C) acting by conjugation and GL(n,C) acting by linear coordinate changes. It can be checked without difficulty from (1) that if ψ2n is GL(n)-basic, then ψr2n is G-basic. We will define the term basic carefully in Section 3.2.2. Given a basic Hochschild 2n-cocycle ψ2n, a Gel’fand–Fuks type procedure (which extends a construction from [10] and [17]) enables one to construct a map of complexes of sheaves fE,ψ2n : ˜hoch(Diff(E))→ Ω2n−• X , where the target has the usual de Rham differential (see Section 3.2 for details). Our extension of the construction from [10, 17] is made possible by a result from [7] (Theorem 5) showing the existence of a “good” choice of Fedosov connection on the bundle BE of fibrewise formal differential operators on E . fE,ψ2n induces a map of complexes fE,ψ2n : Γc(X, ˜hoch(Diff(E)))→ Ω2n−• c (X). The subscript ‘c’ above indicates “compact supports”. As a map of complexes of sheaves, fE,ψ2n depends upon the choice of a Fedosov connection on BE . Nevertheless, as a map in the bounded derived category Db(ShC[X]) of sheaves of C-vector spaces on X, fE,ψ2n is independent of the choice of Fedosov connection on the bundle of fibrewise formal differential operators on E (see Proposition 15). Let τ2n denote the normalized Hochschild cocycle from [11]. Let [· · · ] denote “class in cohomology”. Since HH2n(Diffn) is 1-dimensional, the ratio [ψ2n] [τ2n] makes sense. 1.3. One of the main results in this paper (proven in Section 3) is the following. Theorem 1. If α ∈ Γc(X, ˜hoch(Diff(E))) is a 0-cycle,∫ X fE,ψ2n(α) = (2πi)n [ψ2n] [τ2n] . lim t→∞ str ( α0e−t∆E ) . 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. On Lefschetz Number Formulae for Differential Operators 3 Theorem 2.∫ X fE,ψ2n(D) = (2πi)n [ψ2n] [τ2n] .str(D). Theorem 1 is proven using results from [18] and [19]. If ψ2n is known explicitly, fE(α) can in principle be locally calculated explicitly for α ∈ Γc(X, ˜hoch(Diff(E))). 1.4 In particular, if we put ψ2n to be the normalized Hochschild cocycle τ2n of [11], the 2n-form fE,τ2n(D) was denoted in [10] by χ0(D). We therefore obtain a different proof of the following Lefschetz number theorem of [10] as well as a generalization of this result. Engeli–Felder theorem [10]. If X is compact and D ∈ Γ(X,Diff(E)), str(D) = 1 (2πi)n ∫ X χ0(D). We sketch in a remark Section 3.5 why the map α 7→ ∫ X fE,ψ2n(α) may be viewed as an “integral of ψr2n over X” in some sense. Upto this point, this paper completes our work in [18] and [19] by combining the results therein with an extension of the trace density construction of [10] using a more delicate choice of Fedosov connection on BE . 1.4.1. Let us briefly summarize our approach to Lefschetz number formulae for differential operators in [18, 19] and the current paper. Our approach is to show that the two sides of Theorem 1 coincide upto a constant fact C(ψ2n) independent of X and E such that C(ψ2n) = [ψ2n] [τ2n]C(τ2n). Compared to the approach in [10], the analytic routine part of this step is minimal: beyond proving that α 7→ lim t→∞ str(α0e−t∆E ) vanishes on 0-boundaries (which required only the most basic heat kernel estimates (see [18])), this step requires a few simple, natural arguments from [18, 19] and the current paper. C(τ2n) can then be determined by computing both sides of Theorem 1 for the identity operator on (P1) ×n (see [19]). In particular, the local, more general result of Theorem 1 is an integral part of our approach and the Engeli–Felder formula follows as a direct corollary of this result. As far as this author can see, local results like Theorem 1 do not appear in the approach in [10]. 1.5. We proceed to prove Conjecture 3.3 from [20] in greater generality. In [20], B. Shoikhet starts with a particular 2n + 1 cocycle Ψ2n+1 ∈ C2n+1 Lie (glfin∞ (Diffn),C) called a lifting formula. He then attempts to construct a map of complexes Θ : Clie • ( glfin∞ (Diff(E)),C ) → Ω2n+1−•(X). For any global holomorphic differential operator D on E , he refers to the number ∫ X Θ(E11(D)) as its holomorphic noncommutative residue (whenever X is compact). Since Ψ2n+1 is not GL(n)- basic, this cannot be done as directly as in [20] unless X is complex parallelizable and E is trivial (as per our understanding). One can construct a complex of “completed” Lie chains L̃ie(Diff(E)) of glfin∞ (Diff(E)) (see Section 2.4). This complex is analogous to h̃och(Diff(E)). For a trivial vector bundle E (with bounded geometry) on a parallelizable (not necessarily compact) complex manifold X, we con- struct a map of complexes of sheaves λ(Ψr 2n+1) : L̃ie(Diff(E))[1]→ Ω2n−• X using B. Shoikhet’s lifting formulas. Let α ∈ Γc(X, L̃ie(Diff(E))[1]) be a nontrivial 0-cycle. The component α0 of α of antiholomorphic degree 0 is a finite matrix with entries in Diff0(E). 4 A.C. Ramadoss Theorem 3.∫ X λ(Ψ2n+1)(α) = C. lim t→∞ str ( Tr(α0)e−t∆E ) , 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 [18] and [19]. In Sections 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). We also remark that if we replace Ψ2n+1 by a GL(n)-basic cocycle Θ2n+1 representing the cohomology class [Ψ2n+1], Theorem 3 holds in full generality (i.e., for arbitrary vector bundles with bounded geometry over complex manifolds that are not parallelizable). Indeed, the proof of Lemma 5.2 of [10] can be modified to show the existence of such a GL(n)-basic representative of [Ψ2n+1]. This settles Conjecture 3.3 of [20]. Convention 0. We now state some conventions that stand throughout this paper. (i) The term complex shall be understood to mean a complex with homological grading with one exception that we point out in (iv) below. (ii) The term double complex will mean “complex of complexes”. Let {Cp,q} be a double complex with horizontal and vertical differentials dh and dv respectively. One associates two total complexes Tot⊕(C) and TotΠ(C) with the double complex C, where( Tot⊕(C) ) n := ⊕p+q=nCp,q, ( TotΠ(C) ) n := Πp+q=nCp,q. The differential we use on the shifted total complexes Tot⊕(C)[k] and TotΠ(C)[k] will satisfy d(x) = dh(x) + (−1)\|x\|dv(x) for any homogenous element x of Tot(C)[k]. (iii) The DG-algebras that we work with naturally come with cohomological grading. If A• is a (cohomologically graded) DG-algebra, we automatically view it as a homologically graded algebra by inverting degrees (i.e., elements of An have chain degree −n). The notation A• in this case is still continued in order not to introduce unnatural notation. The same principle holds for DG-modules over DG-algebras that we work with. (iv) When computing the hypercohomology of a complex F• of sheaves on X, we convert F• into a complex with cohomological grading by inverting degrees. (v) For a complex F• of sheaves on X, H•(F) shall denote the (graded) sheaf of homologies of F•. 2 The Hochschild chain complex Γ(X, ˜hoch(Diff(E))) and other preliminaries 2.1. Let Diff•(E)(U) denote Γ(U,Ω0,• X ⊗OX Diff(E)). 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 ⊕kDiff•(E�k)(Uk)[k − 1] where E�k denotes the vector bundle π∗1E ⊗ · · · ⊗ π∗kE on Xk (with πj : Xk → X denoting the projection to the j-th factor). We denote the resulting complex by Ĉ•(Diff•(E)(U)). On Lefschetz Number Formulae for Differential Operators 5 Let ˜hoch(Diff(E)) denote the sheaf associated to the presheaf U 7→ Ĉ•(Diff•(E)(U)) of complexes of C-vector spaces. Note that left multiplication of a homogenous element of Diff•(E)(U) by a smooth function on U is well defined. It follows that each Diff•(E�k)(Uk) is a graded left C∞(U)-module via f.D := π∗1(f).D for all f ∈ C∞(U), D ∈ Diff•(E�k)(Uk). Hence, ˜hoch(Diff(E)) is a complex each of whose terms is a module over the sheaf of smooth functions on X (though the differential is not compatible with this C∞-module structure). It follows that ˜hoch(Diff(E)) is a complex of flabby sheaves on X. Therefore, 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. Proposition 1. Hi(Γ(X, ˜hoch(Diff(E)))) ' H2n−i(X,C) whenever 0 ≤ i ≤ 2n. Proof. Note that the differential on the Hochschild chain complex C•(Γ(U,Diff(E))) extends to a differential on the graded vector space ⊕kΓ(Uk,Diff(E�k))[k − 1]. Call the resulting complex Ĉ•(Γ(U,Diff(E))). One can then consider the sheaf Ĉ•(Diff(E)) associated to the presheaf U 7→ Ĉ•(Γ(U,Diff(E))) of complexes of C-vector spaces. We claim that the natural map of complexes from Ĉ•(Diff(E)) to ˜hoch(Diff(E)) is a quasiisomorphism. To prove our claim, it suffices to verify that the natural map of complexes from Ĉ•(Diff(E)) to ˜hoch(Diff(E)) locally induces an isomorphism on cohomology sheaves. Indeed, on any open subset U of X on which E and T 1,0 X are trivial, the natural map from Γ(Uk,Diff(E�k)) to Γ(Uk,Diff•(E�k)) is a quasiisomorphism (since OXk is quasiisomorphic to (Ω0,• Xk , ∂̄) and Γ(Uk,Diff(E�k)) is a free module over Γ(Uk,OXk)). The desired proposition then follows from (2) and the fact that Ĉ•(Diff(E)) is quasiisomorphic to the shifted constant sheaf C[2n]. This fact (see [18, Lemma 3]) is also implicit in many earlier papers, for instance, [5]. � Convention 1. Whenever we identify Hi(Γ(X, ˜hoch(Diff(E)))) with H2n−i(X,C), we use the identification coming from a specific quasiisomorphism between Ĉ•(Diff(E)) and C[2n]. This quasiisomorphism is constructed as follows. “Morita invariance” yields a quasiisomorphism between Ĉ•(Diff(E)) and Ĉ•(Diff(X)) (see the proof of Lemma 3 of [18]). By a result of [5], Ĉ•(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∈S2n sgn(σ) ( 1, z1, ∂ ∂z1 , . . . , zn, ∂ ∂zn ) to 1[2n] on any open subset U of X with local holomorphic coordinates z1, . . . , zn. We remark that in the above formula, any permutation σ in S2n permutes the last 2n factors leaving the first one fixed. We denote this identification by βE as we did in [18]. 6 A.C. Ramadoss 2.1.1. As in [18], we define a topology on Diff•(E) as follows. Let Diff≤k,•(E) denote Γ(X,Ω0,• X ⊗Diff≤k(E)) where Diff≤k(E) denotes the sheaf of holomorphic differential operators on E of order ≤ k. Equip E and Ω0,• 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 \|\|D\|\|φ,K,s = Sup{\|\|φD(s)(x)\|\| \|x ∈ K}. 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 C∞-connection on E (see Definition 2.4 of [2]) and F ∈ Γ(X,End(E)). Definition 1. We say that E has bounded geometry if for some choice of Hermitian metric on E , there exists a connection OE on E such that ∆E = ∆E +F , where ∆E is the Laplacian of OE and F ∈ Γ(X,End(E)), and all covariant derivatives of the curvature of OE 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 L2 denote the Hilbert space of square integrable sections of K•E. Then, e−t∆E makes sense as an integral operator on K•E L2 for any t > 0 (see [9]). Let Γc denote the functor “sections with compact support”. Suppose that α ∈ Γc(X, ˜hoch(Diff(E))) is a 0-chain. Let α0 denote the component of α with antiholomorphic degree 0. We recall the main results of [18] and [19] in the following theorem. Theorem 4. (1) For any 0-chain α, α0e−t∆E is a trace class operator on K•E L2 for any t > 0. (2) Further, if α is a 0-cycle, lim t→∞ str ( α0e−t∆E ) = 1 (2πi)n ∫ X [α], where [α] denotes the class of α in H2n c (X,C). Remark 1. Let us recall some aspects of [18] and [19]. When X is compact, the linear functional α 7→ lim t→∞ str ( α0e−t∆E ) (3) on Γc(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 Ihoch FLS . To show that the Feigin–Losev–Shoikhet cocycle indeed gives a linear functional on H0(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 [18]). This estimate is proven in “greater than usual” detail in [18]. 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(Uk,Diff•(E�k)). Sheafification then gives On Lefschetz Number Formulae for Differential Operators 7 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 · · · 1−t←−−−− · · · N←−−−− · · · 1−t←−−−− · · ·y y y ˜hoch(Diff(E))2 1−t←−−−− ˜bar(Diff(E))2 N←−−−− ˜hoch(Diff(E))2 1−t←−−−− · · ·y y y ˜hoch(Diff(E))1 1−t←−−−− ˜bar(Diff(E))1 N←−−−− ˜hoch(Diff(E))1 1−t←−−−− · · ·y y y ˜hoch(Diff(E))0 1−t←−−−− ˜bar(Diff(E))0 N←−−−− ˜hoch(Diff(E))0 1−t←−−−− · · ·y y y · · · 1−t←−−−− · · · N←−−−− · · · 1−t←−−−− · · · (4) 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 D1 ⊗ · · · ⊗Dk 7→ (−1)k(−1)dk(d1+···+dk−1)Dk ⊗D1 ⊗ · · · ⊗Dk−1, where the Di are homogenous elements of Γ(U,Diff•(E)) of degree di. t then extends to Γ(Uk,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 + t2 + · · · + tk−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. We omit the proof of the following standard proposition. Proposition 2. Π is a quasi-isomorphism. Applying the linear functional Ihoch FLS on the first column of the double complex of com- pactly supported global sections of the double complex (4), one obtains a linear func- tional ICycl FLS on Γc(X, ˜Cycl(Diff(E))). Moreover, the endomorphism 1 − t of the vector space Γc(X, ˜hoch(Diff(E))0) kills the direct summand Diff0(E). By definition, Ihoch FLS (α) depends only on the component of α in Diff0(E). It follows that Ihoch FLS vanishes on the image of 1 − t. It therefore, induces a linear functional on Γc(X, ˜Co(Diff(E))) which we will denote by ICo FLS. More explicitly, if α is a 0-chain in Γc(X, ˜Co(Diff(E))), ICo FLS(α) = limt→∞ str(α0e−t∆E ) where α0 is the component of α in Diff0(E). The following proposition is a consequence of part 2 of Theorem 4. Proposition 3. ICycl FLS and ICo FLS descend to linear functionals on H0(Γc(X, ˜Cycl(Diff(E)))) and H0(Γc(X, ˜Cycl(Diff(E)))) respectively. Proof. We have already observed that Ihoch FLS vanishes on the image of 1 − t in Γc(X, ˜hoch(Diff(E))0). That it also vanishes on the image of the Hochschild boundary b of Γc(X, ˜hoch(Diff(E))) is implied by part 2 of Theorem 4. The desired proposition then follows from the construction of ICycl FLS and ICo FLS. � 8 A.C. Ramadoss We recall from [19] that H0(Γc(X, ˜Cycl(Diff(E)))) ' H2n(X,C). It follows from Proposition 2 that H0(Γc(X, ˜Co(Diff(E)))) ' H2n(X,C) as well. Denote the natural projection from Γc(X, ˜hoch(Diff(E))) to Γc(X, ˜Co(Diff(E))) by p. The following proposition is immediate from the definition of ICo FLS. Proposition 4. The following diagram commutes. H0(Γc(X, ˜hoch(Diff(E)))) H0(p)−−−−→ H0(Γc(X, ˜Co(Diff(E))))yIhochFLS ICo FLS y C id−−−−→ C Note that if β is a nontrivial 0-cycle of Γc(X, ˜hoch(Diff(E))), the above proposition and Theo- rem 4 together imply that ICo FLS(p(β)) = 1 (2πi)n ∫ X [β] 6= 0. Therefore, ICo FLS is a nontrivial linear functional on H0(Γc(X, ˜Co(Diff(E)))). 2.4. Lie algebra homology. We begin this subsection by reminding the reader of a con- vention 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 {Cp,q := (∧pg)−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. 2.4.1. 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 glr(Diff•(E)) is a sheaf of topological Lie algebras on X with the topology on Γ(U, glr(Diff•(E))) induced by the topology on Γ(U,Diff•(E)) in the obvious way. Treating Γ(U, glfin∞ (Diff•(E))) as the direct limit of the topological vector spaces Γ(U, glr(Diff•(E))) makes glfin∞ (Diff•(E)) a sheaf of topological Lie algebras on X. In this case, the complex of “completed” Lie chains Ĉ lie • (Γ(U, glfin∞ (Diff•(E)))) is obtained by replacing ∧kΓ(U, glfin∞ (Diff•(E))) by the image of the idempotent 1 k! ∑ σ∈Sk sgn(σ)σ acting on glfin∞ (C)⊗k⊗Γ(Uk,Diff•(E�k)). Here, σ acts on the fac- tors glfin∞ (C)⊗k and Γ(Uk,Diff•(E�k)) simultaneously. In an analogous fashion, one can construct the complex Ĉ lie • (Γ(U, glfin∞ (Diff(E)))). Note that Ĉ lie • (Γ(U, glfin∞ (Diff(E)))) is quasi-isomorphic to Ĉ lie • (Γ(U, glfin∞ (Diff•(E)))) since the columns of the corresponding double complexes are quasi- isomorphic. Definition 2. We define L̃ie(Diff(E)) to be the complex of sheaves associated to the complex of presheaves U 7→ Ĉ lie • (Γ(U, glfin∞ (Diff•(E)))) of C-vector spaces on X. In the proposition below, the subscript ‘c’ indicates “compact supports”. On Lefschetz Number Formulae for Differential Operators 9 Proposition 5. Hi(Γc(X, L̃ie(Diff(E)))) ' H−ic (Sym≥1(C[2n+ 1]⊕ C[2n+ 3]⊕ · · · )). Proof. Recall from [12] as well as Section 10.2 of [15] that for any C-algebra A, we have an isomorphism Hlie ≥1(glfin∞ (A)) ' Sym≥1(HC•(A)[1]). Section 10.2 of [15] shows the following facts (i) The complex Clie • (glfin∞ (A))glfin∞ (C) is a commutative and cocommutative DG-Hopf-algebra. (ii) One has a quasiisomorphism from the primitive part Prim(Clie • (glfin∞ (A))glfin∞ (C)) to the Connes complex Cλ •(A)[1]. (iii) The natural map of complexes Clie • (glfin∞ (A))→ Clie • (glfin∞ (A))glfin∞ (C) is a quasi-isomorphism. It follows from these facts (and the Milnor–Moore and PBW theorems) that one has the quasi-isomorphisms Clie • (glfin∞ (A))→ Clie • (glfin∞ (A))glfin∞ (C) ← Sym≥1(Prim(Clie • (glfin∞ (A))glfin∞ (C))), Sym≥1(Prim(Clie • (glfin∞ (A))glfin∞ (C)))→ Sym≥1(Cλ •(A)[1]), Sym≥1(CC•(A)[1])→ Sym≥1(Cλ •(A)[1]). The reader should recall that we follow Convention 2 when defining the Lie chain complex. The last quasi-isomorphism above is induced by the standard quasi-isomorphism from CC•(A) to Cλ •(A). The discussion of Section 10.2 of [15] goes through with the obvious modifications when the algebra A is replaced the sheaf of topological algebrasDiff(E) and Lie algebra and cyclic chains are completed in the sense mentioned above (see Section 2.3.3 of [4] for a similar assertion in the C∞ case). Recall from [22] (see also [23, 18]) that the completed cyclic chain complex of Diff(E) is quasi-isomorphic to C[2n] ⊕ C[2n + 2] ⊕ · · · . It follows that Ĉ lie • (glfin∞ (Diff(E))) is isomorphic to Sym≥1(C[2n + 1] ⊕ C[2n + 3] ⊕ · · · ) in the derived category D(ShC(X)) of sheaves of C-vector spaces on X. Further note that Ĉ lie • (glfin∞ (Diff(E))) is quasi-isomorphic to L̃ie(Diff(E)) (since OX is quasi-isomorphic to Ω0,• X ). Hence, H−ic (L̃ie(Diff(E))) is isomorphic to H−ic (Sym≥1(C[2n+ 1]⊕ C[2n+ 3]⊕ · · · )) for all i. As in Section 2.1, one argues that L̃ie(Diff(E)) is a complex of flabby sheaves on X. Hence, Hi(Γc(X, L̃ie(Diff(E)))) is isomorphic to H−ic (L̃ie(Diff(E))), proving the desired proposition. � Note that Sym≥1(C[2n+1]⊕C[2n+3]⊕· · · ) is a direct sum of copies of the constant sheaf C concentrated in homological degrees ≥ 2n + 1. Let a(n, i) denote the number of partitions of i into pairwise distinct summands from the set {2n+ 1, 2n+ 3, 2n+ 5, . . . }. By Proposition 3.2.2 of [14], Corollary 1. Hi(Γc(X, L̃ie(Diff(E)))) ' ⊕j∈ZHj c(X,C)a(n,i+j) for all i ∈ Z. In particular, H1(Γc(X, L̃ie(Diff(E)))) ' H2n c (X,C) and Hi(Γc(X, L̃ie(Diff(E)))) ' 0 ∀ i ≤ 0. 2.4.2. For any algebra A, we have a map of complexes L : Clie • (glfin∞ (A))[1]→ Cλ •(A), 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 + 1 chain M0 ∧ · · · ∧Mk maps to 10 A.C. Ramadoss the k-chain ∑ σ∈SkM0 ⊗Mσ(1) ⊗ · · · ⊗Mσ(k) of Cλ •(MN (A)) for N sufficiently large. One then applies the generalized trace map to obtain a k chain in Cλ •(A). It is easily verified that the map L extends by continuity to a map of complexes from Ĉ lie • (Γ(U, glfin∞ (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 Γc(X, L̃ie(Diff(E))[1]) to Γc(X, ˜Co(Diff(E))) which we shall continue to denote by L. Proposition 6. L induces a nonzero map from H0(Γc(X, L̃ie(Diff(E))[1])) to H0(Γc(X, ˜Co(Diff(E)))). Proof. Recall from [15] that for any algebra A, L induces an isomorphism between the primitive part of Hlie • (glfin∞ (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 H•(Ĉ lie • (Γ(U, glfin∞ (Diff•(E))))) and the completed cyclic homology of Γ(U,Diff•(E)). In par- ticular, it induces an isomorphism between H2n(Ĉ lie • (Γ(U, glfin∞ (Diff•(E))))[1]) and H2n(Ĉ λ •(Γ(U, glfin∞ (Diff•(E))))) both of which are isomorphic to C. The desired proposition follows from this. � It follows from Propositions 4 and 6 that the map ICo FLS ◦L induces a nonzero linear functional on H0(Γc(X, L̃ie(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 Γc(X, L̃ie(Diff(E))[1]), the component of L(α) in Diff0(E) is Tr(α0) where α0 is the component of α in glfin∞ (Diff0(E)) and Tr is the usual matrix trace. Therefore, I lie FLS(α) = lim t→∞ str ( Tr(α0)e−t∆E ) . Let F be another holomorphic vector bundle on X. There is a natural map ι : Diff•(E) → Diff•(E ⊕ F) of DG-algebras on X. ι induces a map ῑ : Γc(X, L̃ie(Diff(E))[1])→ Γc(X, L̃ie(Diff(E ⊕ F))[1]). Proposition 7. The following diagram commutes H0(Γc(X, L̃ie(Diff(E))[1])) H0(ῑ)−−−−→ H0(Γc(X, L̃ie(Diff(E ⊕ F))[1]))yIlie,EFLS Ilie,E⊕FFLS y C id−−−−→ C Proof. Let α be a 0-chain in Γc(X, L̃ie(Diff(E))[1]). Clearly, Tr(ῑ(α)0) = ι(Tr(α0)). It then follows immediately that lim t→∞ str ( Tr(ῑ(α0))e−t∆E⊕F ) = lim t→∞ str ( Tr(α0)e−t∆E ) 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. � On Lefschetz Number Formulae for Differential Operators 11 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 [10] and [20]. We extend a Gelfand-Fuks type construction from [10] in this section. This construction is a version of a construction that was originally done in [13] by I.M. Gel’fand 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 [10]. 3.1.1. Let r be the rank of E . Let JpE denote the bundle of p-jets of local trivializations of E . In particular, J1E 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 Ex. The group G := GL(n,C) ×GL(r,C) acts on the right on JpE for each p. More specifically, given a local isomorphism of bundles Cn × Cr → E , GL(n,C) acts by linear coordinate transformations on Cn and GL(r,C) acts by linear transformations on Cr. This makes J1E a principal G-bundle. Let J∞E denote lim←− JpE . Since the natural submersions Jp+1E → JpE are G-equivariant, the natural map J∞E → J1E descends to a map J∞E/G→ J1E/G = X. Since the fibres of this map are contractible (see [10]), there exists a smooth section ϕ of J∞E/G over X. This is equivalent to a G-equivariant section of J∞E over J1E . This section is unique upto smooth homotopy. 3.1.2. Let On denote C[[y1, . . . , yn]]. Let Wn denote the Lie algebra of formal holomorphic vector fields (i.e., those of the form n∑ i=1 fi(y1, . . . , yn)∂i with fi ∈ On and ∂i := ∂ ∂yi ). Let Wn,r denote the Lie algebra Wn n glr(On). This semidirect product comes from the action of Wn on gl(On) by derivations. We recall from [10] that there is a map of Lie algebras from Wn,r to the Lie algebra of holomorphic vector fields on J∞E which yields an isomorphism Wn,r → T 1,0 φ J∞E for each φ ∈ J∞E . This is equivalent to a Wn,r-valued holomorphic 1-form Ω on J∞E satisfying the Maurer–Cartan equation dΩ + 1 2 [Ω,Ω] = 0. Since there is a natural map of Lie algebras Wn,r → Mr(Diffn), Ω may be viewed as a Maurer– Cartan form with values in Mr(Diffn). We clarify that for each φ = (φp) a point in J∞E , T 1,0 φ J∞E := lim←−T 1,0 φp JpE and Ω1,0 φ J∞E := lim−→Ω1,0 φp JpE , etc. This is compatible with the definition of J∞E as lim←− JpE . This enables us to make sense of tangent and cotangent spaces of the “infinite dimensional complex manifold” J∞E , and hence, make sense of vector fields and differential forms on J∞E . 3.1.3. Consider the bundle BE := J1E ×GMr(Diffn) of algebras with fiberwise product. Any trivialization of J1E over an open subset U of X yields an isomorphism of algebras C∞(U,BE) ' C∞(U,Mr(Diffn)). (5) More generally, a trivialization of J1E over U yields an isomorphism of graded algebras Ω•(U,BE))→ Ω•(U,Mr(Diffn)). The G equivariant section ϕ : J1E → J∞E allows us to pull back Ω. This gives us a one form ω := ϕ∗Ω ∈ Ω1(J1E ,Mr(Diffn))G. This in turn yields a flat connectionD on the bundle of algebras BE . Let us be explicit here. Given a trivialization of J1E over U , ω descends to a Mr(Diffn) valued one form ωU on U satisfying the Maurer–Cartan equation. The isomorphism (5) identified the connection D with the “twisted” de Rham differential d + [ωU ,−]. More generally, the DG- algebra (Ω•(U,BE), D) is identified with the DG-algebra (Ω•(U,Mr(Diffn)), d + [ωU ,−]). Over any open subset U of X, the degree 0 sections D̂ of Ω•(BE) satisfying DD̂ = 0 12 A.C. Ramadoss 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•(BE) of DG-algebras on X. 3.1.4. Fedosov differentials have earlier been constructed in a holomorphic setting in [16], followed by [6] and more recently, [7]. Construction of a Fedosov differential D on Ω•(BE) has been explained in [7] (see also Dolgushev, [8]) in the case when E = OX . 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 [7]: note that we do not need Part 1 of this result). Theorem 5. There is a map of sheaves of DG-algebras Diff•(E)→ (Ω•(BE), D). Given that the construction of the Fedosov differential D from [7] is a priori different from that of [10], one would like to know whether the Fedosov differential D from [7] can also be obtained via the construction from [10] 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, the next paragraph shows that there exists a G- equivariant section ϕ : J1E → J∞E associated with any Fedosov differential on Ω•(BE). Further, Proposition 8 in Section 3.1.5 shows that for every D ∈ Γ(U,Diff(E)), the section of BE over U that D yields via ϕ (via the construction in Section 3.1.3) coincides with the flat (with respect to D) section of BE corresponding to D. We also point out that on any open subset U of X on which J1E is trivial, any trivialization of J1E over U identifies the DG-algebra (Ω•(U,BE), D) with the DG-algebra (Ω•(U,Mr(Diffn)), d+ [ωU ,−]) for some Mr(Diffn)-valued Maurer–Cartan form even in the construction of [7]. 3.1.5. The argument here follows Section 4.3 of [11]. Consider any point (x, F ) of J1E (F being an extended frame over x). For any holomorphic differential operator D on E over a neighborhood of x, we have a unique section D̂ of Ω(BE) satisfying DD̂ = 0. The frame F identifies D̂ with an element of Mr(Diffn). This yields an isomorphism of algebras Θ : JetsxDiff(E)→ Mr(Diffn). Moreover, since the above isomorphism preserves the order of the differential operator, we obtain an isomorphism θ : JetsxEnd(E)→ Mr(On). This corresponds to an element in the fibre of J∞E over x. We therefore, obtain a section ϕ : J1E → J∞E . It is easy to verify that this section is G-equivariant. Note that the isomorphism θ induces an isomorphism θ̃ from JetsxDiff(E) to Mr(Diffn). Proposition 8. θ̃ coincides with Θ. Proof. Note that Mr(On) and Id⊗Wn generate Mr(Diffn) as an algebra. It therefore suffices to check that θ̃−1 and Θ−1 coincide on Mr(On) and Id⊗Wn.Observe that the image of Wn,r := Wnnglr(On) in Mr(Diffn) is precisely the space of all elements of the form X = D+f for some D ∈ Id⊗Wn and some f ∈ Mr(On). We shall continue to denote this image by Wn,r. Claim. Every derivation of Mr(On) is of the form [X,−] where X ∈ Wn,r. Further, if X ∈ Mr(Diffn) and [X,−] is a derivation of Mr(On), then X ∈Wn,r. We first prove the proposition assuming our claim. Indeed, Θ−1 restricted to Mr(On) coin- cides with θ̃−1 restricted to Mr(On) (both of which coincide with θ−1). For any X ∈Wn,r and for any g ∈ Mr(On), [θ̃−1(X), θ−1(g)] = θ̃−1([X, g]) = θ−1([X, g]) = Θ−1([X, g]) = [Θ−1X, θ−1g]. It follows that for any X ∈ Wn,r, θ̃ −1(X) − Θ−1(X) commutes with all elements of JetsxEnd(E). On Lefschetz Number Formulae for Differential Operators 13 In other words, θ̃−1(X) − Θ−1(X) must be an element of θ−1(id ⊗ On) for all X ∈ Wn,r. In particular, if X ∈ id⊗Wn, then θ̃−1(X) = Θ−1(X) + θ−1(g) for some g ∈ id⊗On. In this case, θ̃−1 ( X2 ) = Θ−1 ( X2 ) + Θ−1(X).θ−1(g) + θ−1(g).Θ−1(X) + θ−1 ( g2 ) = Θ−1 ( X2 ) + Θ−1(2gX) + θ−1 ( [X, g] + g2 ) . It follows that θ̃−1(X2)−Θ−1(X2) is in θ−1(id⊗On) only if g = 0. This shows that θ̃−1 and Θ−1 coincide on id⊗Wn as well. Our claim remains to be proven. Recall (from [15] for instance) that for any algebra A, HH1(A) ' Der(A,A) Inner derivations . In our case, the composite map Wn → HH1(On,On)→ HH1(Mr(On),Mr(On)) 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 Mr(Diffn), there is an element v of Wn such that the image of v under the above composite map coincides with that of φ in HH1(Mr(On),Mr(On)). Consider the derivation v̄ := [id ⊗ v,−] of Mr(On). It is easy to verify that the image of φ − v̄ in HH1(Mr(On),Mr(On)) is zero. Hence, φ − v̄ is an inner derivation. In other words, as derivations on Mr(On), φ = [id⊗ v + f,−] for some v ∈ Wn and some f ∈ Mr(On). This proves the first part of our claim. The second part of our claim is then easy: if [X,−] is a derivation of Mr(On), then [X,−] = [Y,−] as derivations on Mr(On) for some Y ∈ Wn,r. Hence, X − Y commutes with every element in Mr(On). This shows that X − Y is an element of id⊗On. � 3.2. A Gel’fand–Fuks type construction. We now recall and extend a construction from [10]. 3.2.1. Let A be any DG-algebra. Let ω ∈ A1 be a Maurer–Cartan element (i.e., an satisfying dω + ω2 = 0). Let Aω denote the DGA whose underlying graded algebra is A but whose differential is d + [ω,−]. Let C•(A) denote the complex of normalized Hochschild chains of A. Let (ω)k denote the normalized Hochschild 0-chain (1, ω, . . . , ω) (k copies of ω). Denote the set of (p, q)-shuffles by Shp.q. There is a shuffle product × : A⊗A⊗p ⊗A⊗A⊗q → A⊗A⊗p+q, (a0, a1, . . . , ap)⊗ (b0, ap+1, . . . , ap+q) 7→ ∑ σ∈Shp,q sgn(σ)(a0b0, aσ−1(1), . . . , aσ−1(p), aσ−1(p+1), . . . , aσ−1(p+q)). Note that C•(A) = ⊕k≥1A⊗ A⊗k[k − 1] as graded vector spaces. The differential on C•(A) extends naturally to a differential on CΠ • (A) := ∏ k≥1 A⊗ A⊗k[k − 1] making the natural map of graded vector spaces a map of complexes. One has the following proposition due to Engeli and Felder [10]. Though [10] used normalized Hochschild chains, their proof goes through in the current context as well. Proposition 9. The map CΠ • (Aω)→ CΠ • (A), x 7→ x× ∑ k (−1)k(ω)k is a term by term isomorphism of complexes. 14 A.C. Ramadoss 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 Diffn. The derivative of this action embeds the Lie algebra gln in Diffn as the Lie subalgebra of operators of the form ∑ j,k aj,kyk∂j . We say that the normalized cocycle α ∈ Cp(Diffn) is GL(n,C)-basic if α is GL(n,C)-invariant and if p∑ i=1 (−1)i+1α(a0, . . . , ai−1, a, ai+1, . . . , ap) = 0 for any a ∈ gln and any a0, . . . , ap ∈ Diffn. More generally, G := GL(n,C) × GL(r,C) acts on Mr(Diffn). The action of GL(n,C) on Mr(Diffn) is induced by the action of GL(n,C) on Diffn and GL(r,C) acts on Mr(Diffn) by conjugation. The derivative of this action embeds g := gln⊕ glr as a Lie subalgebra of Mr(Diffn) of elements of the form ∑ j,k aj,kzk∂j ⊗ idr×r +B where B ∈ Mr(C). Again, a cocycle α in Cp(Mr(Diffn)) is said to be G basic if α is G-invariant and p∑ i=1 (−1)i+1α(a0, . . . , ai−1, a, ai+1, . . . , ap) = 0 for any a ∈ g and any a0, . . . , ap ∈ Mr(Diffn). The following proposition is immediate from equation (1). Proposition 10. If ψ2n is GL(n,C)-basic then ψr2n is G-basic. We also recall that a cocycle α ∈ Cp(Mr(Diffn)) is said to be continuous if it depends only on finitely many Taylor coefficients of its arguments. 3.2.3. Let ψr2n be a continuous, G-basic Hochschild 2n-cocycle of Mr(Diffn). Evaluation at ψr2n gives a map of graded vector spaces CΠ • (Ω•(U,Mr(Diffn)), d)→ Ω2n−•(U), (ω0a0, . . . , ωpap) 7→ ψr2n(a0, . . . , ap)ω1 ∧ · · · ∧ ωp for all a1, . . . , ap ∈ Mr(Diffn), ω1, . . . , ωp ∈ Ω•(U). (6) By convention, ψr2n(a0, . . . , ap) = 0 if p 6= 2n. Proposition 11. The map (6) is a map of complexes (with Ω2n−p(U) equipped with the diffe- rential (−1)pdDR). Proof. Note that any Hochschild p-chain α of A := (Ω•(U,Mr(Diffn)), d) can be expressed as a sum of terms ∑ i αi where αi has de Rham degree i and is an element of A⊗Ap+i. By definition, ψr2n(α) = ψr2n(α2n−p). Let δ denote the differential on the Hochschild chain complex of A. Then, (δα)2n−p = (−1)pdDRα2n−p +dhochα2n−p+1. Since ψr2n is a cocycle, ψr2n(dhochα2n−p+1) = 0. The desired proposition is now immediate. � Observe that there is a natural map of complexes C•(Ω •(U,BE), D) → CΠ • (Ω•(U,BE), D). Also, the graded involution on Ω•(U) acting on Ωp(U) by multiplication by (−1)b p 2 c is a map of complexes between (Ω2n−•(U), (−1)•dDR) and Ω2n−•(U). Recall from Section 3.1.4 that a trivialization of J1E over U identifies (Ω•(U,BE), D) with Ω•ω(U,Mr(Diffn)) where ω is a Mr(Diffn)-valued 1-form on U satisfying the Maurer–Cartan equation. By Propositions 9 and 11, we have a map of complexes C•(Ω •(U,BE), D)→ Ω2n−•(U). (7) Explicitly, the map (7) maps a chain µ ∈ Cp(Ω •(U,BE), D) to (−1)b 2n−p 2 cψr2n ( µ̂× ∑ k (−1) k (ω)k ) , where µ̂ is the image of µ in CΠ p (Ω•ω(U,Mr(Diffn))). Proposition 12. The map (7) is independent of the trivialization of J1E used. On Lefschetz Number Formulae for Differential Operators 15 Proof. A different trivialization of J1E over U differs from the chosen one by a gauge change g : U → G. A section µ of Ω•(U,BE) transforms into g.µ. The Maurer–Cartan form ω is replaced by g.ω−dg.g−1. That ψr2n(α× ∑ k±(ω)k) = ψr2n(g.α× ∑ k±(g.ω−dg.g−1)k) is immediate from the fact that ψr2n is G-basic. This proves the desired proposition. � It follows that the map (7) gives a map of complexes of presheaves( U 7→ C•(Ω •(U,BE), D) ) → ( U 7→ Ω2n−•(U) ) . (8) 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 (8) depends on the Fedosov diffe- rential D. It follows from Theorem 5 that we obtain a composite map of complexes of presheaves( U 7→ C•(Diff•(E)(U)) ) → ( U 7→ C•(Ω •(U,BE), D) ) → ( U 7→ Ω2n−•(U) ) . (9) Proposition 13. The composite map (9) extends to a map of complexes of sheaves ˜hoch(Diff(E))→ Ω2n−• X . Proof. Denote the composite map (9) by fE . For each open subset U of X, fE yields a map of complexes from C•(Diff•(E)(U)) to Ω2n−•(U). As described in Section 2.1.1, Ĉ•(Diff•(E)(U)) has the structure of a (graded) locally convex topological vector space. One can verify with- out difficulty that the subcomplex C•(Diff•(E)(U)) is dense in Ĉ•(Diff•(E)(U)) and that the differential of Ĉ•(Diff•(E)(U)) is continuous. Similarly, Ω2n−•(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 fE is continuous. Then, fE extends to a map of complexes from Ĉ•(Diff•(E)(U)) to Ω2n−•(U) for each U . This can be easily see to yield a map of complexes of presheaves( U 7→ Ĉ•(Diff•(E)(U)) ) → ( U 7→ Ω2n−•(U) ) . On sheafification, f yields a map of complexes of sheaves ˜hoch(Diff(E))→ Ω2n−• X proving the desired proposition. Indeed, continuity of fE follows from continuity of ψr2n. Since ψr2n is continuous, C l-norms of f(a0 ⊗ · · · ⊗ ak) over a compact subsets of U are estimated by C l ′ -norms of finitely many Taylor coefficients of the images âi of the ai in Ω•(U,BE) over compact subsets of U . � The map of complexes of sheaves obtained in Proposition 13 depends on the cocycle ψ2n. We shall denote this map by fE,ψ2n. Recall that there is a natural map of complexes of sheaves Ĉ•(Diff(E)) → ˜hoch(Diff(E)). Let U be a subset of X with local holomorphic coordinates z1, . . . , zn such that E is trivial over U . By Convention 1, the standard generator for the homology of Γ(U, Ĉ•(Diff(E))) is a cycle cE(U) mapping to the normalized Hochschild 2n-cycle∑ σ∈S2n sgn(σ)σ ( IdE ⊗ 1, IdE ⊗ z1, IdE ⊗ ∂ ∂z1 , . . . , IdE ⊗ zn, IdE ⊗ ∂ ∂zn ) . Let cr2n denote the normalized Hochschild 2n-cycle∑ σ∈S2n sgn(σ)σ ( Idr×r ⊗ 1, Idr×r ⊗ y1, Idr×r ⊗ ∂ ∂y1 , . . . , Idr×r ⊗ yn, Idr×r ⊗ ∂ ∂yn ) of Mr(Diffn). Note that ψr2n(cr2n) makes sense since ψr2n is G-basic. 16 A.C. Ramadoss Proposition 14. fE,ψ2n(cE(U)) = ψr2n(cr2n). Proof. Note that any elementD ofDiff(E)(U) gives a holomorphic Mr(Diffn)-valued function D̃ on J∞E\|U , satisfying dD̃ + [Ω, D̃] = 0 where Ω is the Maurer–Cartan form from Section 3.1.2. It follows that there is a map of complexes C•(Diff(E)(U))→ C•((Ω •,0(J∞E\|U ,Mr(Diffn)), d+ [Ω,−])) → CΠ • (Ω•,0(J∞E\|U ,Mr(Diffn))). The last arrow is by Proposition 9. Evaluation at ψr2n followed by applying the involution multiplying p-forms by (−1)b p 2 c therefore yields a map of complexes C•(Diff(E)(U))→ Ω2n−•,0(J∞E\|U ). Given a section ϕ : J1E → J∞E and a trivialization of J1E over U , one has a composite map C•(Diff(E)(U))→ Ω2n−•,0(J∞E\|U )→ Ω2n−•(J1E\|U )→ Ω2n−•(U). (10) The middle arrow above is pullback by the section ϕ : J1E → J∞E . The rightmost arrow above is pullback by the section of J1E arising out of the trivialization of J1E that we have chosen over U . Any section ϕ : J1E\|U → J∞E\|U is unique upto homotopy. It follows from this that the composite map (10) is unique upto homotopy for a fixed trivialization of J1E over U . This fact and the proof of Proposition 12 together imply that the image of cE(U) under (10) is independent of the precise choice of ϕ and of trivialization of J1E over U . To compute it, we could choose ϕ to be the section taking ∂ ∂zi to the formal derivative ∂ ∂yi and zi to yi + ai at (a1, . . . , an). In this situation, the image of cE(U) is indeed seen to be ψr2n(cr2n) in homological degree 2n. For any section ϕ of J∞E\|U over J1E and for any trivialization of J1E over U , the map (10) takes (D0, . . . ,D2n) to ψr2n(D̂0, . . . , D̂n) where D̂ is the section of Ω•(U,Mr(Diffn)) corresponding to D (satisfying dD̂ + [ω, D̂] = 0 where ω is as in Section 3.1.3). Let ϕ be a section of J∞E\|U over J1E associated with the Fedosov differential D on Ω•(BE) as in Section 3.1.5. Proposition 8 implies that fE,ψ2n takes (D0, . . . ,D2n) to ψr2n(D̂0, . . . , D̂n). Thus, fE,ψ2n(cE(U)) = ψr2n(cr2n). � 3.3. Properties of fE,ψ2n. Let Db(ShC[X]) denote the bounded derived category of sheaves of C-vector spaces on X. Then, the map fE,ψ2n of Proposition 13 induces a morphism in Db(ShC[X]). Recall that the natural embedding of C into Ω•(U) for any open U ⊂ X is a quasi-isomorphism. This induces a quasiisomorphism of complexes of sheaves C→ Ω•X on X between the constant sheaf C and Ω•X . Proposition 15. The following diagram commutes in Db(ShC[X]) ˜hoch(Diff(E)) fE−−−−→ Ω2n−• Xxβ−1 E x C[2n] −−−−→ [ψ2n] [τ2n] id C[2n] In particular, the map fE in Db(ShC[X]) is independent of the choice of Fedosov connection on BE . Proof. As objects in Db(ShC[X]), ˜hoch(Diff(E)) as well as Ω2n−• 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), HomDb(ShC[X])(C,C) ' C. On Lefschetz Number Formulae for Differential Operators 17 It follows that it is enough to verify this proposition for each open subset U of X with local holo- morphic coordinates on which E is trivial. For such a U , the generalized trace map maps cE(U) to a 2n-cycle mapping to the normalized Hochschild 2n-cycle r ∑ σ∈S2n sgn(σ)σ ( 1, z1, ∂ ∂z1 , . . . , zn, ∂ ∂zn ) . It follows from Convention 1, Section 2.1 that βE(cE(U)) = r. Therefore, 1 r cE(U) is a 2n-cycle mapped to 1 by βE . It follows that fE(β −1 E (1)) = 1 rfE(cE(U)) = 1 rψ r 2n(cr2n). The last equality is by Proposition 14. However ψr2n(cr2n) = [ψr2n] [τr2n] τ r 2n(cr2n). But [ψr2n] [τr2n] = [ψ2n] [τ2n] by equation (1). Further, since τ2n(c2n) = 1 (see [11]), τ r2n(cr2n) = r by (1). Therefore, 1 rψ r 2n(cr2n) = [ψ2n] [τ2n] . This proves the desired proposition. � Since fE,ψ2n is a map of complexes of sheaves, the map Γc(X, ˜hoch(Diff(E))→ C, α 7→ ∫ X fE,ψ2n(α) (11) descends to a (nonzero) linear functional on H0(Γc(X, ˜hoch(Diff(E)))). We shall denote this linear functional by ∫ X fE,ψ2n . 3.4. Proof of Theorem 1. Proposition 15 implies that [fE,ψ2n(α)] = [ψ2n] [τ2n] [α] in H2n c (X,C) for any 0-cycle α in Γc(X, ˜hoch(Diff(E)). Theorem 1 follows immediately from this observation and Theorem 4. 3.5. Remarks. fE,ψ2n induces a map of complexes C•(Diff•(E))→ Ω2n−•(X). This can equivalently be viewed as a family Θi ∈ Ωi(C2n−i(Diff•(E))) of cochain valued forms satisfying the differential equations dDRΘi = ±δΘi+1, where δ is the differential on the Hochschild cochain complex C2n−i(Diff•(E)). In particular, Θ2n is a 2n-form with values in C0(Diff•(E)). When X is compact, ∫ X Θ2n is precisely the Hochschild 0-cocycle (11). This viewpoint seeing (11) as coming from “integrating ψr2n over X” is taken by [20]. 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 ν2n+2p (see [17, 22]). One obtains a map of complexes Γc(X, ˜Cycl(Diff(E)))→ Ω2n+2p−• c (X) as a result. When X is compact, and k ≥ p, the above map on the 2k-th homology yields a map H2n−2k(X,C)⊕H2n−2k+2(X,C)⊕ · · · ⊕H2n(X,C)→ H2n+2p−2k(X,C) It would be interesting to understand this map further. 18 A.C. Ramadoss 4 Integrating the lifting formula Ψ2n+1 We begin by outlining the construction of Ψ2n+1 in Section 4.1. Section 4.2 is devoted to proving Theorem 3. Notation. For any algebra A, Mr(A) will denote the algebra of r × r matrices with entries in A. glfin∞ (A) will denote the Lie algebra of finite matrices with entries in A. M∞(A) will denote the algebra of infinite matrices M with entries in A such that the (i, j)-th entry of M vanishes whenever \|i− j\| > C(M) where C(M) is a constant (depending on M). 4.1. About the lifting formula Ψ2n+1. 4.1.1. The only explicit fact about the Lifting formula Ψ2n+1 that we shall use in this paper is its form. Let ΨDiffn denote the algebra C[[y1, . . . , yn]][y−1 1 , . . . , y−1 n ][[∂−1 1 , . . . , ∂−1 n ]][∂1, . . . , ∂n] (∂i := ∂ ∂yi ) of formal pseudodifferential operators. The coefficient at y−1 1 · · · y−1 n ∂−1 1 · · · ∂−1 n defines a linear functional on ΨDiffn. This functional vanishes on commutators of elements of ΨDiffn. We therefore call it a trace and denote it by TrΨDiffn : ΨDiffn → C (see [1]). Recall that the “usual” matrix trace Trglfin∞ yields a linear map from glfin∞ (A) to A for any algebra A. It can then be verified that TrΨDiffn ◦ Trglfin∞ yields a trace on the algebra glfin∞ (ΨDiffn). For any α ∈ A, let α ⊗ 1∞ denote the diagonal matrix α ∈ M∞(A). Let Di, 1 ≤ i ≤ 2n be the derivations on glfin∞ (ΨDiffn) given by Di = ad(ln(xi) ⊗ 1∞) for 1 ≤ i ≤ n and Di = ad(ln(∂i−n)⊗ 1∞) for n+ 1 ≤ i ≤ 2n. We recall from [21] that [Di, Dj ] = ad(Qij ⊗ 1∞) for some elements Qij of ΨDiffn. 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 i1, . . . , ik. Define O(t)(A1, . . . , A2n+1) = AltA,D(TrΨDiffn ◦ Trglfin∞ (P1,t ◦ · · · ◦ P2n+1,t)), where if j is marked, Pj,t = AjQj,j+1 and Pj+1,t = Aj+1, Pj,t = Dj(Aj) if j and j − 1 are not marked and j 6= 2n+ 1, P2n+1,t = A2n+1. Note that 0 and 2n should be thought of as unmarked by default in the above formula. If t is the “empty” marking, O(t)(A1, . . . , A2n+1) = AltA,D(TrΨDiffn ◦ Trglfin∞ (D1(A1) · · ·D2n(A2n)A2n+1)). Theorem 6 ([21]). The linear functional (A1, . . . , A2n+1) 7→ ∑ t∈S O(t)(A1, . . . , A2n+1) is a 2n+ 1-cocycle in C2n+1 Lie (glfin∞ (ΨDiffn);C). The natural inclusion of algebras Diffn ⊂ ΨDiffn extends to an inclusion glfin∞ (Diffn) ⊂ glfin∞ (ΨDiffn). The formula from Theorem 6 therefore, gives us the cocycle Ψ2n+1 ∈ C2n+1 Lie (glfin∞ (Diffn),C). Ψ2n+1 is referred to as a lifting formula. On Lefschetz Number Formulae for Differential Operators 19 4.1.2. Note that for any algebra A, we have isomorphisms Mm(A)⊗Mr(C) ' Mrm(A) of algebras. Taking the direct limit of these isomorphisms, we obtain a map of algebras (and therefore, Lie algebras) im : glfin∞ (Mm(A)) ' Mm(A)⊗ glfin∞ (C)→ glfin∞ (A). It follows that Ψ2n+1 yields a cocycle in C2n+1 lie (glfin∞ (Mm(Diffn)),C) as well. We denote this cocycle by Ψm 2n+1. For M ∈ Mm(Diffn), M ⊗ id shall refer to the element im(M ⊗ id) of M∞(Diffn). For any p > m, let ιm,p : Mm(Diffn) → Mp(Diffn) denote the natural embedding obtained by “padding with zeros”. Proposition 16. If A1, . . . , A2n,M ∈ Mm(Diffn) and B1, . . . , B2n ∈ Mp(Diffn) then Ψm+p 2n+1((A1 ⊕B1)⊗ id, . . . , (A2n ⊕B2n)⊗ id, ιm,m+p(M)⊗ E1,1) = Ψm 2n+1(A1 ⊗ id, . . . , A2n ⊗ id,M ⊗ E1,1). Proof. It is enough to show that for any t ∈ S, O(t)((A1 ⊕B1)⊗ id, . . . , (A2n ⊕B2n)⊗ id, ιm,m+p(M)⊗ E1,1) = O(t)(A1 ⊗ id, . . . , A2n ⊗ id,M ⊗ E1,1). (12) The summands on the left hand side of (12) are of the form TrΨDiffn ◦Trglfin∞ (X1 ◦ · · · ◦Xr) where Xi = (Aj ⊕Bj)⊗ id or Xi = ιm,m+p(M)⊗E1,1 or Xi = λ⊗ id for some element λ of ΨDiffn. It is easy to see that this summand does not change if each Xi of the form Xi = (Aj ⊕ Bj) ⊗ id is replaced by Aj ⊗ id. Doing this however transforms the sum on the left hand side to that on the right hand side. � 4.2. Constructing Shoikhet’s holomorphic noncommutative residue in general. 4.2.1. Let g be a DG-Lie algebra. Then, {Cp,q := (∧pg)−q\|p ≥ 1} becomes a double complex whose horizontal differential is the Chevalley–Eilenberg differential and whose vertical diffe- rential is that induced by the differential intrinsic to g. We denote the complex Tot⊕(C•,•) by Clie • (g). Similarly we denote TotΠ(C•,•) by CΠ,lie • (g). There is a natural map of complexes from Clie • (g) to CΠ,lie • (g). Let ω ∈ g1 be a Maurer–Cartan element. Let gω denote the twisted Lie algebra whose underlying differential is d+ [ω,−]. Proposition 17. There is a natural map of complexes CΠ,lie • (gω)→ CΠ,lie • (g), (g0, . . . , gk) 7→ ∑ j≥0 1 j! (g0, . . . , gk, ω, ω, . . . , ω) (j ω’s). Proof. Since this proposition is completely analogous to Proposition 2.4 of [10], we shall only sketch the proof. Denote the differential of g by d. Let dCE denote the Chevalley–Eilenberg differential. Let (ω)j := ω ∧ · · · ∧ ω j times. Step 1. One first notes that dCE(ω)j = j(j − 1) 2 [ω, ω] ∧ (ω)j−2 = −j(j − 1)dω ∧ (ω)j−2 = −jd ( ωj−1 ) . 20 A.C. Ramadoss The middle equality above is because ω is a Maurer–Cartan element. It follows that if φj := 1 j!(ω)j , then dCEφj = −dφj−1. (13) Step 2. Let g0, . . . , gk be homogenous elements of g. Let di denote the degree of gi. Let G := g0 ∧ · · · ∧ gk. One then verifies (by a direct calculation) that dCE(G ∧ φj) = (dCEG) ∧ φj + (−1)k+1G ∧ dCEφj + (−1)d0+···+dk+k+1(ad(ω)G) ∧ φk−1. (14) The desired proposition follows from equations (13) and (14) 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 Clie • (g) and CΠ,lie • (g) by Clie • (g, d) and CΠ,lie • (g, d) respectively. 4.2.2. Let D be the Fedosov differential on Ω•(BE) chosen as in Section 3.1.4. This choice ensures that there is a morphism Diff•(E)→ Ω•(BE) of sheaves of DG-algebras on X. Let U be an open subset of X on which J1E is trivial. Recall that any trivialization of J1E on an open subset U of X induces an term by term isomorphism of sheaves DG-algebras (Ω•(BE)U , D) → (Ω•U (Mr(Diffn)), d+[ω,−]). Also, ω⊗1N is a Maurer–Cartan element in glfin∞ (Ω•(U,Mr(Diffn))) for any sufficiently large natural number N . Here 1N denotes the N×N identity matrix “padded with 0’s” on the right and bottom to obtain an element of glfin∞ (C). We will continue to denote this element by ω for notational brevity. One therefore has the following composite map of complexes Clie • (glfin∞ (Ω•(U,BE), D)) −−−−→ Clie • (glfin∞ (Ω•(U,Mr(Diffn)), d+ [ω,−])) θ y y CΠ,lie • (glfin∞ (Ω•(U,Mr(Diffn)), d)) ←−−−− CΠ,lie • (glfin∞ (Ω•(U,Mr(Diffn)), d+ [ω,−])) (15) The horizontal arrow on top is from the isomorphism of (Ω•(U,BE),D) with (Ω•(U,Mr(Diffn)), d+ [ω,−]) induced by ϕ. The vertical arrow on the right is the natural map mentioned in Sec- tion 4.2.1. The horizontal arrow on the bottom is from Proposition 17. Let Ξ2n+1 be any continuous 2n+1 cocycle in C2n+1 lie (glfin∞ (Mr(Diffn)),C). As in Proposition 11, Section 3.2.3, eva- luation at Ξ2n+1 yields a map of complexes from CΠ,lie • (glfin∞ (Ω•(U,Mr(Diffn)), d))[1] to Ω2n−•(U) (with the differential on Ω2n−p being (−1)pdDR). Composing this map with θ, and applying the involution multiplying a p-form by (−1)b p 2 c, we obtain a map of complexes (ϕ,Ξ2n+1)∗ : Clie • (glfin∞ (Ω•(U,BE), D))[1]→ Ω2n−•(U). Explicitly, if µ ∈ Clie p (glfin∞ (Ω•(U,BE), D)), (ϕ,Ξ2n+1)∗(µ) = (−1)b 2n−p 2 c∑ k 1 k! Ξ2n+1(µ̂ ∧ ωk), (16) where µ̂ is the image of µ in Clie p (glfin∞ (Ω•(U,Mr(Diffn)), d+ [ω,−])). In particular, we obtain a map of complexes Clie • (Γ(U, glfin∞ (Diff•(E))))[1]→ Ω2n−•(U). Proposition 18. The above map extends to a map λ(Ξ2n+1) : L̃ie(Diff(E))[1]→ Ω2n−• U of complexes of sheaves of C-vector spaces on U . On Lefschetz Number Formulae for Differential Operators 21 Proof. This follows from the continuity of Ξ2n+1. The argument proving this is completely analogous to the proof of Proposition 13. However, since Ξ2n+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 . � Corollary 2. If X is complex parallelizable and E is trivial, λ(Ξ2n+1) yields a map of complexes of sheaves L̃ie(Diff(E))[1]→ Ω2n−• X on X. Proof. The obstruction to globalizing λ(Ξ2n+1) comes from the fact that there is no consistent way of choosing a section of J1E over X in general. � Remark 2. If Ξ2n+1 is GL(n) × GL(r)-basic, λ(Ξ2n+1) yields a map of complexes of sheaves L̃ie(Diff(E))[1] → Ω2n−• X for arbitrary (not necessarily complex parallelizable) X and arbitrary (not necessarily trivial) E . The proof of this assertion is completely analogous to that of Propo- sition 12. 4.2.3. Properties of λ(Ξ2n+1). Proposition 19. For a fixed trivialization of J1E, the map induced by λ(Ξ2n+1) in Db(ShC[U ]) depends only on the class of Ξ2n+1 in H2n+1 lie (Mr(Diffn)). Proof. We will show that if Ξ2n+1 is a coboundary, then (ϕ,Ξ2n+1)∗ is null-homotopic. The homotopy we shall provide will automatically yield a homotopy between λ(Ξ2n+1) and 0. Recall that the map (ϕ,Ξ2n+1)∗ was obtained by composing the map θ from the commu- tative diagram (15) with evaluation at Ξ2n+1 followed by multiplying p-forms by (−1)b p 2 c. If Ξ2n+1 = dβ, evaluation at β gives a homotopy between evaluation at Ξ2n+1 and 0. This proves that (ϕ,Ξ2n+1)∗ is null-homotopic when Ξ2n+1 = dβ. Explicitly, if µ ∈ Clie p (glfin∞ (Ω•(U,BE), D)), our homotopy maps µ to (−1)b 2n−1−p 2 c∑ k 1 k!β(µ̂ ∧ ωk) where µ̂ is the image of µ in Clie p (glfin∞ (Ω•(U,Mr(Diffn)), d+ [ω,−])). It is easy to see that since β is a continuous cochain, our homotopy yields a homotopy between λ(dβ) and the 0 map. � Proposition 20. The map induced by λ(Ξ2n+1) in Db(ShC[U ]) is independent of the trivializa- tion of J1E chosen. Proof. Let h be a Lie subalgebra of g. Let H•lie(g, h) denote the Lie algebra cohomology of g relative to h (with coefficients in the trivial module). The proof of Lemma 5.2 of [11] goes through with minor modifications to show that H2n+1 lie (glfin∞ (Mr(Diffn)), glfin∞ ⊕ gln ⊕ glr) ' H2n+1 lie (glfin∞ (Mr(Diffn))). It follows that there exists a G-basic cocycle Θ2n+1 such that [Θ2n+1] = [Ξ2n+1] in H2n+1 lie (glfin∞ (Mr(Diffn))). By Proposition 19, λ(Ξ2n+1) = λ(Θ2n+1) as a morphism in Db(ShC[U ]). The proof that λ(Θ2n+1) is independent of the chosen trivialization of J1E (as a morphism in Db(ShC[U ])) is identical that of Proposition 12 (with the obvious modifications). � Proposition 21. The map induced by λ(Ξ2n+1) in Db(ShC[U ]) is independent of the Fedosov differential D on Ω•(BE) (for a fixed trivialization of J1E over U). Proof. From the discussion in Section 2.4.1, L̃ie(Diff(E))[1] is isomorphic to C[2n] ⊕ V in Db(ShC[U ]) where V is a direct sum of shifted constant sheaves concentrated in homological degree ≥ 2n + 2. Also, Ω2n−• is isomorphic to C[2n] in Db(ShC[U ]). Since C is an injective object in the category of sheaves of C-vector spaces, HomDb(ShC[U ])(V,C[2n]) = 0. It therefore, suffices to show that λ(Ξ2n+1) applied to a fixed 2n cycle generating the 2n-st homology of Γ(U, L̃ie(Diff(E))[1]) is independent of the choice of Fedosov differential. 22 A.C. Ramadoss In fact, since J1E is trivial over U , the 2n-th homology of Γ(U, L̃ie(Diff(E))[1]) is in fact generated by a 2n-cycle clie E (U) in Ĉ lie • (Γ(U, glfin∞ (Diff(E))))[1] (see Section 2.4.1). As in the proof of Proposition 14, given any (G-equivariant) section ϕ : J1E → J∞E , one has a composite map of complexes Ĉ lie • (Γ(U, glfin∞ (Diff(E))))[1]→ Ω2n−•,0(J∞E\|U )→ Ω2n−•(J1E)→ Ω2n−•(U). (17) The “restriction” of the first map above to Clie • (Γ(U, glfin∞ (Diff(E))))[1] is constructed just like the analogous map in the proof of Proposition 14. The middle arrow is ϕ∗ where ϕ : J1E → J∞E . The third is the pullback by the chosen section of J1E . Since ϕ is unique upto homotopy, the image of clie E (U) under the composite map (17) is independent of ϕ. Finally, Proposition 8 tells us that if ϕ is the section associated with the Fedosov connection D on BE (as in Section 3.1.5), the image of clie E (U) under the composite map (17) coincides with the image of clie E (U) under the “restriction” of λ(Ξ2n+1) to Ĉ lie • (Γ(U, glfin∞ (Diff(E))))[1]. This proves the desired proposition. � Propositions 20 and 21 show that the map induced by λ(Ξ2n+1) in Db(ShC[U ]) is independent of the choices made to define it. Let F be another holomorphic vector bundle on X. Let ι denote the natural map of sheaves of DG-algebras between Diff•(E) and Diff•(E ⊕ F). Note that ι induces a map of sheaves ῑ : L̃ie(Diff(E))[1]→ L̃ie(Diff(E ⊕ F))[1] on X. Let r and s be the ranks of E and F respectively. Recall that for every positive integer m we have the lifting cocycle Ψm 2n+1 described in Section 4.1.2 (Ψ1 2n+1 = Ψ2n+1). Proposition 22. The following diagram commutes in Db(ShC[U ]) L̃ie(Diff(E))[1] ῑ−−−−→ L̃ie(Diff(E ⊕ F))[1]yλ(Ψr2n) λ(ψr+s2n ) y Ω2n−• U id−−−−→ Ω2n−• U Proof. Recall that there is a natural map of complexes from Ĉ lie • (Γ(U, glfin∞ (Diff(E))))[1] to Γ(U, L̃ie(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 Ĉ lie • (Γ(U,Diff(E)))[1] is isomorphic to L̃ie(Diff(E))[1] in Db(ShC[U ]). It therefore, suffices to show that the following diagram commutes upto homology Ĉ lie • (Γ(U, glfin∞ (Diff(E))))[1] ῑ−−−−→ Ĉ lie • (Γ(U, glfin∞ (Diff(E ⊕ F))))[1]yλ(Ψr2n) λ(ψr+s2n ) y Ω2n−• U id−−−−→ Ω2n−• U (18) By arguments paralleling the proofs of Propositions 20 and 21, we are free to choose the sections U → J∞E\|U and U → J∞(E ⊕ F)\|U that we shall use. Fix holomorphic coordinates z1, . . . , zn on U . Fix a trivialization of E over U . Also fix a trivialization of F over U . If {ei} and {fj} are the ordered bases of Γ(U, E) and Γ(U,F) (over Γ(U,O)) we have chosen in our trivi- alizations, we trivialize E⊕F by choosing the ordered basis {e1, . . . , er, f1, . . . , fs} of Γ(U, E ⊕ F). With these choices, we obtain sections of J1E and J1(E ⊕ F) over U . The chosen coordinate system and trivialization of E identify Γ(U,Diff(E)) with Mr(Diff(U)). The (G-equivariant) section of J∞E we choose is the one that maps ∂ ∂zi to ∂ ∂yi and f(z1, . . . , zn) to (a1, . . . , an) 7→ ∑ I ∂f ∂zI ∣∣∣ (a1,...,an) yI . On Lefschetz Number Formulae for Differential Operators 23 LetD ∈ Γ(U, glfin∞ (Diff(E))) be arbitrary. Let D̂ denote the flat section of C∞(U, glfin∞ (Mr(Diffn))) corresponding to D. Note that by our choices of sections of J∞E and J∞(E ⊕ F), ι̂(D) = ιr,r+sD̂. It follows from equation (16) and Proposition 16 that the following diagram commutes literally (not just upto homology) Clie • (Γ(U, glfin∞ (Diff(E))))[1] ῑ−−−−→ Clie • (Γ(U, glfin∞ (Diff(E ⊕ F))))[1]yλ(Ψr2n) λ(ψr+s2n ) y Ω2n−• U id−−−−→ Ω2n−• U (19) Since Clie • (Γ(U, glfin∞ (Diff(E))))[1] is dense in Ĉ lie • (Γ(U, glfin∞ (Diff(E))))[1] (and similarly for E⊕F), and since all the maps involved in diagram (19) are continuous, commutativity of diagram (18) follows (note that this “commutativity on the nose” of diagram (18) is with our convenient choice of section of J∞E\|U over J1E\|U as well as our choice trivialization of J1E 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. Further, if Ξ2n+1 is replaced by a GL(n)×GL(r)-basic cocycle having the same class in cohomology, Proposition 20 is replaced by the obvious analog of Proposition 12, which goes through for arbitrary X and arbitrary E with bounded geometry. Similarly, if Ψ2n+1 is replaced by a GL(n)-basic cocycle representing the same cohomology class, Propositions 21 and 22 hold for arbitrary X and arbitrary E and F with bounded geometry. 4.2.4. Proof of Theorem 3. For this subsection, we shall assume that X is complex parallelizable, E is trivial and E has bounded geometry. The first two assumptions can be re- moved if Ψ2n+1 is replaced by a GL(n)-basic cocycle representing the same cohomology class. By Propositions 20 and 21, for any choice used in the construction of λ(Ψr 2n+1), the map∫ X λ(Ψr 2n+1) : Γc(X, L̃ie(Diff(E))[1])→ C, α 7→ ∫ X λ(Ψr 2n+1)(α) induces the same map on the 0-th homology of Γc(X, L̃ie(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 Γc(X, L̃ie(Diff(E))[1]) as well. By Corollary 1, H0(Γc(X, L̃ie(Diff(E))[1])) is a 1-dimensional C-vector space. It follows that as linear functionals on H0(Γc(X, L̃ie(Diff(E))[1])),∫ X λ(Ψr 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 = OX . Let U be an open subset of X with holomorphic coordinates that identify U with an open disk in Cn. Choose a nontrivial 0-cycle α of Γc(U, L̃ie(Diff(E))[1]). Note that after making the necessary choices, the construction of λ(Ψ2n+1) is local in nature. It follows that∫ U λ(Ψ2n+1)(α) = ∫ X λ(Ψ2n+1)(j∗α), (20) where j : U → X is the natural inclusion. 24 A.C. Ramadoss Further, if φ is an element of Diff0(X) that is compactly supported on U , str ( φe−t∆X ) = str ( φe−t∆U ) (21) 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•OU. To see this, note that if pt(x, y) is the kernel of e−t∆X , str(φe−t∆X ) = ∫ X str(φpt(x, x))\|dx\| = ∫ U str(φpt(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•OU is fixed (see [9]). It follows from equation (20) (in the case when E = OX) and equation (21) that C(X, E) is independent of X as well. This proves Theorem 3. 4.2.5. For this subsection, let X be an arbitrary compact smooth manifold. Let F• be a complex of sheaves on X such that each Fi is a module over the sheaf of smooth functions on X. Suppose also that F• is in Db(ShC[X]). Let U := {Ui} be a finite good cover of X. Consider the double complex Č(U,F•) where Čp,q(U,F•) = ⊕j1<···<jpΓc(Uj1 ∩ · · · ∩ Ujp ,Fq). Here, Γc denotes “sections with compact support”. The vertical differential in this double complex is induced by the differential on F•. The horizontal differential δ is given by the formula (δω)j1,...,jp−1 = N∑ j0=1 (ω)j0,...,jp−1 . Here, (α)i1,...,ik denotes the component of a chain α ∈ Čk,l(U,F•) in Γc(Ui1 ∩ · · · ∩ Uik ,Fl). We also follow the convention that if i1 < · · · < ik, (α)iσ(1),...,iσ(k) = sgn(σ)(α)i1,...,ik for any permuta- tion σ of 1, . . . , k. The following proposition is a trivial modification of Proposition 12.12 of Bott and Tu [3]. We therefore, omit its proof. Proposition 23. (1) The map Tot•(Č(U,F•))→ Γc(X,F•), α 7→ ∑ i (α)i is a quasi-isomorphism. (2) Hp(Tot•(Č(U,F•))) ' H−pc (X,F•). 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. Let α be 0-cycle in Γc(X,F•). Proposition 24. Suppose that F• is acyclic in negative degrees. If X is compact, there exist 0-cycles αi of Γc(Ui,F•) such that ∑ i[αi] = [α] in H0(Γc(X,F•)). Proof. We use a “staircase argument”. By Proposition 23, there exists a 0-cycle α̂ of Tot•(Č(U,F•)) such that [α̂] maps to [α] under the quasi-isomorphism in Proposition 23. Note that α̂ = ∑ k≥1 α̂k where α̂j is a −k + 1-chain in ⊕j1<···<jkΓc(Uj1 ∩ · · · ∩ Ujk ,F•). Let N(α̂) On Lefschetz Number Formulae for Differential Operators 25 be the largest integer such that α̂N(α̂) 6= 0. Then, α̂N(α̂) is a cycle in ⊕j1<···<jkΓc(Uj1 ∩ · · · ∩ UjN(α̂) ,F•). Since, F• is acyclic in negative degrees, α̂N(α̂) = δβ for some β where δ is the differential on ⊕j1<···<jkΓc(Uj1 ∩ · · · ∩ UjN(α̂) ,F•). Let d denote the differential in the bi- complex Tot•(Č(U,F•)). View β as a chain in Tot•(Č(U,F•)). Then, α̂ + dβ is homologous to α̂. On the other hand, N(α̂ + dβ) = N(α̂) − 1. It follows that repeating this procedure eventually gives a 0-cycle in ⊕iΓc(Ui,F•) homologous to α̂. This proves the desired proposi- tion. � 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 := {Ui} be a finite good cover of X. Let D be a global holomorphic differential operator on E . Note that E11(D) is a 0-cycle in Γc(X, L̃ie(Diff(E))[1]). Applying Proposition 24 to the complex L̃ie(Diff(E))[1] of sheaves on X, we see that there exist 0-cycles αi in Γc(Ui, L̃ie(Diff(E))[1]) such that ∑ i[αi] = [E11(D)] in H0(Γc(X, L̃ie(Diff(E))[1])). Define the holomorphic noncommutative residue of D to be the sum NC(D) := ∑ i ∫ Ui λ(Ψr 2n+1)(αi). One can make the choices used to define λ(Ψr 2n+1) in the Ui arbitrarily. We also remark that λ(Ψr 2n+1) is not globally defined on X in a direct way (at least as far as we can see). By Theorem 3, NC(D) = ∑ i ∫ Ui λ(Ψr 2n+1)(αi) = C. ∑ i I lie,E FLS (αi) = C.I lie,E FLS (∑ i αi ) = C.I lie,E FLS (E11(D)) = C. lim t→∞ str ( De−t∆E ) = C.str(D). This proves Conjecture 3.3 of [20] in greater generality. Alternatively, one can sidestep the complications that arose while defining NC(D) as follows. Let Θ2n+1 be a GL(n)-basic cocycle representing the cohomology class [Ψ2n+1] (its existence is proven following the proof of Lemma 5.2 in [10]). Construct the cocycles Θr 2n+1 from Θ2n+1 exactly as the Ψr 2n+1 were constructed from Ψ2n+1. The cocycles Θr 2n+1 are GL(n) × GL(r)- basic representatives of the class [Ψr 2n+1]. Then, λ(Θr 2n+1) is a globally defined 2n-form and define NC(D) := ∫ X λ(Θr 2n+1)(E11(D)). The two definitions of NC(D) given here are equivalent by Proposition 19. 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. I thank the referees of this article for their constructive suggestions. This work was done (prior to my joining my current position) partly at Cornell University and partly at IHES. I am grateful to both these institutions for providing me with a congenial work atmosphere. 26 A.C. Ramadoss References [1] Adler M., On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg–de Vries type equations, Invent. Math. 50 (1979), 219–248. [2] Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Springer-Verlag, Berlin, 2004. [3] Bott R., Tu L.W., Differential forms in algebraic topology, Springer-Verlag, Berlin, 1995. [4] Bressler P., Kapranov M., Tsygan B., Vasserot E., Riemann–Roch for real varieties, in Algebra, Arithmetic and Geometry: in Honor of Yu.I. Manin, Progr. Math., Vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, 125–164, math.DG/0612410. [5] Brylinski J.-L., A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), 93–114. [6] Calaque D., Dolgushev V., Halbout G., Formality theorems for Hochschild chains in the Lie algebroid setting, J. Reine Angew. Math. 612 (2007), 81–127, math.KT/0504372. [7] Calaque D., Rossi C., Lectures on Duflo isomorphisms in Lie algebras and complex geometry, Lecture notes, available at http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf. [8] Dolgushev V., Covariant and equivariant formality theorems, Adv. Math. 191 (2005), 147–177, math.QA/0307212. [9] Donelly H., Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), 485–496. [10] Engeli M., Felder G., A Riemann–Roch–Hirzebruch formula for traces of differential operators, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 621–653, math.QA/0702461. [11] Feigin B.L., Felder G., Shoikhet B., Hochschild cohomology of the Weyl algebra and traces in deformation quantization, Duke Math. J. 127 (2005), 487–517, math.QA/0311303. [12] Feigin B.L., Tsygan B.L., Cohomology of the Lie algebra of generalized Jacobian matrices, Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 86–87 (in Russian). [13] Gel’fand I.M., Fuks D.B., Cohomology of the Lie algebra of tangent vector fields on a smooth manifold. II, Funktsional. Anal. i Prilozhen. 4 (1970), no. 2, 23–31 (in Russian). [14] Kashiwara M., Schapira P., Sheaves on manifolds, Springer-Verlag, Berlin, 2002. [15] Loday J.-L., Cyclic homology, Springer-Verlag, Berlin, 1992. [16] Nest R., Tsygan B., Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems, Asian J. Math. 5 (2001), 599–635, math.QA/9906020. [17] Pflaum M.J., Posthuma H.B., Tang X., An algebraic index theorem for orbifolds, Adv. Math. 210 (2007), 83–121, math.KT/0507546. [18] Ramadoss A.C., Some notes on the Feigin–Losev–Shoikhet integral conjecture, J. Noncommut. Geom. 2 (2008), 405–448, math.QA/0612298. [19] Ramadoss A.C., Integration over complex manifolds via Hochschild homology, J. Noncommut. Geom. 3 (2009), 27–45, arXiv:0707.4528. [20] Shoikhet B., Integration of the lifting formulas and the cyclic homology of the algebra of differential opera- tors, Geom. Funct. Anal. 11 (2001), 1096–1124, math.QA/9809037. [21] Shoikhet B., Lifting formulas. II, Math. Res. Lett. 6 (1999), 323–334, math.QA/9801116. [22] Willwacher T., Cyclic cohomology of the Weyl algebra, arxiv:0804.2812. [23] Wodzicki M., Cyclic homology of differential operators, Duke Math. J. 54 (1987), 641–647. http://dx.doi.org/10.1007/BF01410079 http://dx.doi.org/10.1007/978-0-8176-4745-2_4 http://arxiv.org/abs/math.DG/0612410 http://dx.doi.org/10.1515/CRELLE.2007.085 http://arxiv.org/abs/math.KT/0504372 http://math.univ-lyon1.fr/~calaque/LectureNotes/LectETH.pdf http://dx.doi.org/10.1016/j.aim.2004.02.001 http://arxiv.org/abs/math.QA/0307212 http://arxiv.org/abs/math.QA/0702461 http://dx.doi.org/10.1215/S0012-7094-04-12733-2 http://arxiv.org/abs/math.QA/0311303 http://arxiv.org/abs/math.QA/9906020 http://dx.doi.org/10.1016/j.aim.2006.05.018 http://arxiv.org/abs/math.KT/0507546 http://dx.doi.org/10.4171/JNCG/25 http://arxiv.org/abs/math.QA/0612298 http://dx.doi.org/10.4171/JNCG/29 http://arxiv.org/abs/0707.4528 http://dx.doi.org/10.1007/s00039-001-8225-5 http://arxiv.org/abs/math.QA/9809037 http://arxiv.org/abs/math.QA/9801116 http://arxiv.org/abs/0804.2812 http://dx.doi.org/10.1215/S0012-7094-87-05426-3 1 Introduction 2 The Hochschild chain complex (X,hoch(Diff(E))"0365hoch(Diff(E))) and other preliminaries 3 Differential forms computing the Lefschetz number 4 Integrating the lifting formula 2n+1 References

Integration of Cocycles and Lefschetz Number Formulae for Differential Operators

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