Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations
aa r X i v : . [ m a t h . K T ] J u l THE SHEAF OF TWISTED CHEREDNIK ALGEBRAS AS A UNIVERSAL FILTERED FORMALDEFORMATION
ALEXANDER VITANOVAbstract. The notion of sheaves of (twisted) global Cherednik algebras ℋ t,c,ψ,X,G attached to a global quotient orbifold X/G was introduced and first studied by Pavel Etingof in . For the special case of an affine variety X with a finitegroup G acting on it, he proved that when t = 1 and c, ψ are formal parameters, ℋ ,c,ψ,X,G is a universal formaldeformation of the smash product 𝒟 X ⋊ G on X/G of the sheaf of differential operators 𝒟 X on X and the group G . Themain result of this note is a generalization of Etingof’s proof to the case of a global quotient orbifold X/G , where X is ageneric smooth analytic or algebraic variety. We prove that in these cases ℋ ,c,ψ,X,G with formal parameters c and ψ isa universal filtered formal deformation of 𝒟 X ⋊ G .First, we construct quasi-isomorphisms between the Hochschild (co)chain complex of 𝒟 X ⋊ G and the G -invariant part ofthe direct sum of sheaves of holomorphic de Rham differential forms on cotangent bundles of the fixed point submanifoldsin X associated to cyclic subgroups of G . By means of them we compute the hypercohomology of the Hochschild (co)chaincomplex of 𝒟 X ⋊ G . Combining these quasi-isomorphisms with results from the theory of filtered algebraic extensions ofsheaves of filtered associative algebras we compute the space 𝒟ℯ𝒻 ( 𝒟 X ⋊ G ) f of isomorphism classes of filtered infinitesimaldeformations of 𝒟 X ⋊ G . Finally, we compute the dimension of 𝒟ℯ𝒻 ( 𝒟 X ⋊ G ) f and consequently prove that the sheafof twisted global Cherednik algebras on X/G is a universal filtered deformation of 𝒟 X ⋊ G in the global case. Contents1.
Introduction 11.1. Outline of the main results 21.2. Notation 32. Deformation Theory of Sheaves of Filtered Associative Algebras 32.1. Square-zero extensions of associative algebras and infinitesimal deformations 32.2. Square-zero extensions of sheaves of filtered algebras and infinitesimal deformations 53. Universal deformation of 𝒟 X ⋊ G 𝒟 X ⋊ G
1. Introduction
In the pioneering work [Eti04] Pavel Etingof proposed a broad generalization of the notion of a rational Cherednikalgebra associated to a complex representation h of a finite group G to a smooth algebraic or analytic variety X withan action by a finite group G of automorphisms. He introduced a sheaf of twisted Cherednik algebras ℋ t,c,ψ,X,G onthe quotient orbifold X/G , where t is a complex number, c are class functions on the set of complex reflections in G and ψ ∈ H ( X, Ω ≥ X ) G , and showed that much of the standard theory of rational Cherednik algebras possessesa natural generalization to the global case. For example, one can derive a global notion of a "Calogero-Moser space"for X by means of the center of the sheaf of Cherednik algebras. From the point of view of deformation theorythe central result in his work is [Eti04, Theorem 2.23] according to which the sheaf of twisted Cherednik algebras ℋ ,c,ψ,X,G with formal parameters c and ψ on X/G , where X is a complex affine variety, is a universal formaldeformation of the skew-group algebra 𝒟 X ⋊ G . Etingof’s proof relies solely on the fact that affine varieties are also D -affine thanks to which the space of infinitesimal deformation of 𝒟 X ⋊ G is isomorphic to the second Hochschildcohomology group of the associative algebra of global sections Γ( X, 𝒟 X ⋊ G ) . This reduces the global statement o the well-known linear case. Unfortunately, as smooth analytic and algebraic varieties are not D -affine in general,one cannot identify 𝒟 X ⋊ G with Γ( X, 𝒟 X ⋊ G ) . That is the reason why the correct version of [Eti04, Theorem2.23] in the non-affine global context demands a fully sheaf-theoretic approach. Even though some version of [Eti04,Theorem 2.23] in the case of generic smooth algebraic and analytic varieties has always been expected to be true untilnow, a proper generalization of that result and a corresponding proof thereof have lacked. The current note closesthis longstanding gap providing necessary mathematical tools along with a final proof of the fact that the sheaf oftwisted Cherednik algebras arises as a universal formal filtered deformation of 𝒟 X ⋊ G in the global algebraic andanalytic case. We briefly elaborate on the main results of the note in the ensuing subsection.1.1. Outline of the main results.
Section 2 is devoted to the definition of the space
𝒟ℯ𝒻 (Λ) of first-order defor-mations of a sheaf of filtered associative algebras Λ . The naive identification of this space with the set exal(Λ , Λ) of locally trivial square-zero extensions of Λ by the Λ − Λ bimodule Λ is not meaningful for the purposes of defor-mation theory because, as noted to the author by Pavel Etingof and Valery Lunts, in many crucial cases such as forinstance Λ = 𝒟 X on elliptic curve X the set exal(Λ , Λ) becomes infinite-dimensional. For sheaves of algebras withinfinite dimensional space of infinitesimal formal deformations it is not clear how to define a meaningful notion of auniversal formal deformation. However, it turns out that in the special case of sheaves of filtered associative algebrasone can restrict itself to a subspace exal( ℳ , Λ) f of filtered square-zero extensions which in many cases includingthe relevant for our aims case Λ = 𝒟 X ⋊ G has the advantage of being finite-dimensional. The main result inthis section is Theorem 2.10 whose proof is an imitation of [Gra61, Theorem 3.2] for the case of filtered Hochschildcochains. Theorem (A).
There is an isomorphism of C -vector spaces between exal f (Λ , ℳ ) and the second hypercohomologygroup H ( X, σ ≥ b 𝒞 • f (Λ , ℳ )) of the filtered Hochschild cochain complex b 𝒞 • f (Λ , ℳ ) of Λ with values in ℳ brutallytruncated at . Section 3 is the core of the paper and it is mostly devoted to the explicit computation of the hypercohomologygroup H ( X, σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G )) in Theorem (A). Subsection 3.1 is by and large a rehash of the well-knowntheory of Calabi-Yau algebras. Here we contribute a definition of sheaves of Calabi-Yau algebras. The main resultis Proposition 3.5 in which we give an explicit locally free resolution of Sym • ( 𝒯 X ) in terms of left Sym • ( 𝒯 X ) e -modules which we use to prove that 𝒟 X and 𝒟 X ⋊ G are sheaves of Calabi-Yau algebras of dimension X ) ( cf. Proposition 3.6 and Corollary 3.7) and later to prove that the canonical inclusion of the complex of filtered cochainsof 𝒟 X ⋊ G in the Hochschild cochain complex of 𝒟 X ⋊ G is a quasi-isomorphism ( cf. Proposition 3.13).In Subsection 3.2 we begin by defining a quasi-isomorphism from the Hochschild chain complex of 𝒟 X ⋊ G tothe G -invariant part of the direct sum of holomorphic de Rham complexes of cotangent bundles, shifted by thedimension of the total space of the bundle, over the strata in X , associated to the cyclic subgroups in G . The quasi-isomorphism is an adaptation of a similar construction in [Vit20] to the holomorphic setup involving trace densitymorphisms for ℋ ,c, ,X,G . The construction of this quasi-isomorphism relies on a technique, developed in [Vit19],which realizes the sheaf of untwisted Cherednik algebras ℋ ,c, ,X,G ( c is not necessarily formal) in terms of gluingof special sheaves π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 Hn − l,l ) of flat sections of algebra bundles over connected components of fixedpoint submanifolds X Hi restricted to open and dense subsets X iH , called strata of X associated to H in G . Thispresentation of the sheaf of untwisted Cherdnik algebras yields a natural projection map from ℋ ,c, ,X,G to thesheaf π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 Hn − l,l ) for every subgroup H of G by means of which we can use an Engeli-Felder typecontsruction [EF08] along the lines of [RT12] and later [Vit20] to define morphism χ Hi : 𝒞 • ( 𝒟 X ⋊ G ) → ( j Hi ∗ π Hi ∗ Ω n − l iH −• T ∗ X Hi ) G where Ω • T ∗ X Hi is the complex of holomorphic differential forms on the cotangent bundle π Hi : T ∗ X Hi → X Hi tothe connected component X Hi with codimension l iH of the fixed point set X H and j Hi is the canonical inclusion of X Hi in X . Using the Calabi-Yau property of 𝒟 X ⋊ G , proven earlier, we arrive at a similar map for the Hochschildcochain complex of 𝒟 X ⋊ G 𝒳 Hi : 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) → ( j Hi ∗ π Hi ∗ Ω •− l iH T ∗ X Hi ) G . Let h g i be the cyclic subgroup of G generated by the element g in G . One of the central results in the whole note isthe following theorem ( cf. Theorem 3.11 and Corollary 3.12) heorem (B). For every choice of a linearly independent trace φ i h g i of b 𝒟 l ig ⋊ h g i and for every family of holomorphicMaurer-Cartan forms { ω α } on the cotangent bundle T ∗ X gi of the connected component of the fixed point submanifold X gi with codimension l ig with values in 𝒜 h g i n − l ig ,l ig the maps ⊕ i,g ∈ G χ gi : b 𝒞 • ( 𝒟 X ⋊ G ) → (cid:16) ⊕ i,g ∈ G j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi (cid:17) G (1) ⊕ i,g ∈ G 𝒳 gi : b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) → (cid:16) ⊕ i,g ∈ G j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi (cid:17) G (2) are quasi-isomorphisms. Moreover, the induced morphisms at the level of homology and cohomology sheaves are canon-ical. By means of quasi-isomorphism (2) from Theorem (B) we compute in Corollary 3.15 the space of infinitesimalfiltered formal deformations
𝒟ℯ𝒻 ( 𝒟 X ⋊ G ) f of 𝒟 X ⋊ G . With this at hand we finally arrive at the main theorem3.16 of the paper. In this note we show how to prove the ensuing result for analytic varieties but all of the presentedresults can be proven in the case of smooth algebraic varieties in a similar fashion. Theorem (Main Theorem).
Let X be a smooth algebraic variety or a smooth analytic variety equipped with a finitesubgroup G ⊂ Aut( X ) acting faithfully on X . The sheaf of twisted Cherednik algebras ℋ ,c,ψ,X,G on the quotientorbifold X/G is a universal formal filtered deformation of the skew-group algebra 𝒟 X ⋊ G . Notation.
From now on untill the end of this paper X will denote a complex analytic manifold of dimension dim C X = n and equipped with an action by a finite group G of holomorphic automorphisms. A complex reflec-tion is an element g in G such that the fixed point submanifold X g has a connected component of codimension .Throughout the note we do not insist that G contains complex reflections.
2. Deformation Theory of Sheaves of Filtered Associative Algebras
First, we quickly review well-known aspects of the theory of extension of associative algebras and its relation toformal deformations.2.1.
Square-zero extensions of associative algebras and infinitesimal deformations.
Let k be a field of char-acteristic zero. In the following we denote by Λ an associative k -algebra with a unit and by Λ op the opposite k -algebra. Let Λ e := Λ ⊗ k Λ op denote the enveloping algebra of Λ . In the ensuing we use that every left Λ e -moduleis a left Λ e -module. Definition 2.1.
An extension of Λ is a k -algebra Γ together with an k -algebra epimorphism σ : Γ → Λ . An extension σ : Γ → Λ is called square-zero extension of Λ if the kernel of σ , ker σ , satisfies ker( σ ) = 0 . In thatcase the left Γ e -module ker( σ ) acquires a well-defined structure of a left Λ e -module. Definition 2.2.
Let A be a left Λ e -module. A square-zero extension of Λ by A is a short exact sequence of k -vectorspaces ζ : 0 → A χ −→ Γ σ −→ Λ → such that Γ is a k -algebra, σ : Γ → Λ is a square-zero extension of Λ and for A , considered as a left Γ e -module bypullback along σ , the k -vector space isomorphism χ : A → ker( σ ) is also an isomorphism of Γ e -modules. Two square-zero extensions ζ : 0 → A χ −→ Γ σ −→ Λ → and ζ ′ : 0 → A χ ′ −→ Γ ′ σ ′ −→ Λ → of Λ by a Λ e -bimodule A are said to be equivalent if there is a k -algebra morphism β : Γ → Γ ′ such that the diagram Γ0 A Λ 0Γ ′ β σχ χ ′ σ ′ commutes. By the -Lemma, if such a morphism β exists, it is an isomorphism of algebras. A square-zero extension ζ : 0 → A χ −→ Γ σ −→ Λ → is k - split if there is a k -linear map θ : Λ → Γ such that σ ◦ θ = id Λ . The splitting map θ defines a k -linear isomorphism Λ ⊕ A ∼ = Γ . The pull-back of the multiplication in Γ to Λ ⊕ A yields a product on he direct sum given by ( λ , a )( λ , a ) = ( λ λ , λ a + a λ + µ ( λ , λ )) for all ( λ , a ) , ( λ , a ) ∈ Γ , where µ ( λ , λ ) = θ ( λ λ ) − θ ( λ ) θ ( λ ) is a Hochschild -cocycle in HH (Λ , A ) called the factor set of the k -split square-zero extension. The so-defined product gives an isomorphism of k -algebras Γ → Λ ⊕ A defining an equivalence ofsquare-zero extensions. A square-zero extension of Λ by A is called split if there is a k -algebra morphism θ : Λ → Γ such that σ ◦ θ = id Λ . In particular, every split extension is k -split. All split extensions of Λ by a left Λ e -module A are equivalent to the semi-direct sum → A → Λ ⊕ A → Λ where Λ ⊕ A is a k -algebra with respect to the product ( λ , a )( λ , a ) = ( λ λ , λ a + a λ ) . The set of iso-morphism classes of extensions of Λ by A is denoted by Ext k (Λ , A ) . The Braer sum of short exact sequences gives Ext k (Λ , A ) the structure of a k -vector space with zero element the isomorphism class of split extensions of Λ by A .Let t be a central element in Λ . We recall the definition of a first-order deformation of Λ . Definition 2.3 (Definition 3.10, [DMZ07]).
Let R be an augmented unital ring with an augmentation ǫ : R → k . A R -deformation of Λ is an associative R -algebra B together with a k -algebra isomorphism B ⊗ k R ∼ = Λ . In this note we are primarily interested in infinitesimal ( first-order ) deformations which are k [ t ] / ( t ) -deformationsin the sense of the above definition. These deformations can be equivalently characterized using Hochschild coho-mology. Proposition 2.4.
Given a k [ t ] / ( t ) -deformation B of Λ , there is a unique Hochschild 2-cocycle µ ∈ Hom k (Λ ⊗ k Λ , Λ) such that the k [ t ] / ( t ) - vector space Λ ⊗ k k [ t ] / ( t ) , equipped with the product λ ∗ λ = λ λ + µ ( λ , λ ) t mod ( t ) ,becomes a k [ t ] / ( t ) -algebra isomorphic to B as k [ t ] / ( t ) -algebras.Proof. Proof is verbatim identical to the proof of Theorem . in [DMZ07]. (cid:3) Two first-order deformations B and B ′ of Λ are said to be equivalent if the coresponding unique Hochschild 2-cocycles µ and µ ′ are cohomologous, e.g. [ µ ] = [ µ ′ ] in HH (Λ , Λ) . Denote by Def(Λ) the set of equivalence classesof infinitesimal deformations B of Λ . We have the following easy to prove but crucial identity. Lemma 2.5.
There is a one-to-one correspondence between the sets
Ext k (Λ , Λ) and Def(Λ) .Proof.
Suppose ζ : 0 → Λ χ −→ Γ σ −→ Λ → is a representative of an isomorphism class in Ext k (Λ , Λ) . Since perassumption k is a field, ζ is k -split. That means that it is equivalent to a square-zero extension ζ : 0 → Λ t χ −→ Λ ⊕ Λ t σ −→ Λ → where we accounted that Λ ∼ = Λ t as Λ e -bimodules. Consequently, for all λ , λ ∈ Λ we have ( λ , λ ,
0) = ( λ λ , µ ( λ , λ )) where µ is a Hochschild -cocycle factor set. By Proposition 2.4, the factor set µ determines a unique k [ t ] / ( t ) -algebra structure on Λ ⊕ Λ t ∼ = Λ ⊗ k k [ t ] / ( t ) which is an infinitesimal deformation of Λ . Any other representativeof the isomorphism class of ζ is equivalent to ζ and thus induces a cohomologous factor set. Hence, there is awell-defined map(3) Ext k (Λ , Λ) → Def(Λ) . Conversely, assume that B is a k [ t ] / ( t ) -deformation of Λ . Again by Proposition 2.4 it is uniquely described by aHochschild -cocycle µ ∈ Hom k (Λ ⊗ , Λ) which according to Theorem 3.1 in [Mac75] corresponds to the factor setof a square-zero extension ζ of Λ by Λ . An equivalent k [ t ] / ( t ) -deformation B ′ of Λ is per definition given by acohomologous Hochschild -cocycle µ ′ ∈ Hom k (Λ ⊗ , Λ) which in turn defines a zero-square extension of Λ by Λ which is equivalent to ζ . Thus, there is a well-defined map Def(Λ) → Ext k (Λ , Λ) such that it and the map (3) are inverses of each other. (cid:3) The bijectivity between
Ext k (Λ , Λ) and Def(Λ) equips the later with the structure of a k -vectors space. .2. Square-zero extensions of sheaves of filtered algebras and infinitesimal deformations.
The material inthe following section is mostly inspired by the content in [Gra61]. Let Λ be a sheaf of associative C -algebras on a(complex) manifold X an let 𝒜 be a left Λ e -module. Definition 2.6.
A square-zero extension of Λ by 𝒜 is a sheaf of C -algebras Γ together with an exact sequence of sheavesof C -vector spaces (4) → 𝒜 i −→ Γ p −→ Λ → in which p is a square-zero extension of sheaves of C -algebras as in Definition 2.1 satisfying the properties of Definition2.2. Two algebra extensions Γ and Γ ′ of Λ by a Λ e -module 𝒜 are said to be equivalent if there is a homomorphism ofsheaves of C -algebras k : Γ → Γ ′ such that the diagram Γ0 𝒜 Λ 0Γ ′ k pi i ′ p ′ commutes. In that case, k is an isomorphism of sheaves of C -algebras by the -Lemma. Unlike the case of square-zeroextensions of associative k -algebras, discussed in the previous section, in general not every zero-square extensionof a sheaf of algebras Λ by a sheaf of Λ e -module splits as a short exact sequence of sheaves of C -vector spaces.Such extensions are of no interest for the study of deformation theory because local infinitesimal deformations ofthe product of a sheaf of algebras always arise from (local) splitting morphisms of sheaves of vector spaces. That iswhy we shall restrict our attention to a special subclass of so-called locally trivial square-zero extensions of sheavesof algebras. Definition 2.7.
A square-zero extension → 𝒜 χ −→ Γ σ −→ Λ → of Λ by 𝒜 is called locally trivial if there is an opencover { U α } of X such that for each U α , the short exact sequence of sheaves of C -vector spaces → 𝒜 | U α χ −→ Γ | U α σ −→ Λ | U α → is C X | U α -split. i.e. there exists a C X -linear homomorphism j α : Λ | U α → Γ | U α such that σ ◦ j α = id Λ | Uα . A square-zero extension → 𝒜 i −→ Γ p −→ Λ → of Λ by 𝒜 is split if it admits a morphism of sheaves of algebras j : Λ → Γ such that p ◦ j = id Γ . In particular, a split square-zero extension is locally trivial. We denote the setof isomorphism classes of locally-trivial square-zero algebra extensions of Λ by 𝒜 by exal(Λ , 𝒜 ) . It is a C -vectorspace with respect to the Baer sum and the isomorphism class of split square-zero extensions of Λ by 𝒜 as the zeroelement.For the study of twisted differential operators, it makes sense to restrict our attention to the subcategory of sheavesof filtered associative C -algebras. In the remainder of this section Λ will be equipped with an increasing filtration Λ ⊆ Λ ⊆ · · · ⊆ Λ n ⊆ . . . which is exhaustive, i.e. Λ = ∪ ∞ i =0 Λ i . Similarly, the Λ e -module 𝒜 will be endowed withan increasing filtration 𝒜 ⊆ 𝒜 ⊆ · · · ⊆ 𝒜 n ⊆ . . . with Λ m · 𝒜 ⊆ 𝒜 m + n which is exhaustive, i.e. 𝒜 = ∪ ∞ i =0 𝒜 i . Definition 2.8.
A filtered square-zero extension of Λ by 𝒜 is a square-zero extension (5) → 𝒜 i −→ Γ p −→ Λ → such that Γ is a sheaf of C -algebras with an increasing exhaustive filtration and i and p are filtered morphisms of C -vector spaces. Definition 2.9.
A filtered square-zero extension → 𝒜 χ −→ Γ σ −→ Λ → of Λ by 𝒜 is locally trivial if there is an opencover { U α } of X such that for each U α , the short exact sequence → 𝒜 | U α χ −→ Γ | U α σ −→ Λ | U α → has a filtration-preserving k -linear homomorphism j α : Λ | U α → Γ | U α such that σ ◦ j α = id Λ | Uα . filtered square-zero extension → 𝒜 i −→ Γ p −→ Λ → of Λ by 𝒜 is split if it admits a morphism of sheavesof C -algebras j : Λ → Γ such that p ◦ j = id Γ . In particular, a split filtered square-zero extension is locally trivial.Two filtered square-zero algebra extensions Γ and Γ ′ of Λ by a Λ e -module 𝒜 are said to be equivalent if there is afiltered homomorphism of sheaves of C -algebras k : Γ → Γ ′ such that the diagram Γ0 𝒜 Λ 0Γ ′ k pi i ′ p ′ commutes. The subspace of exal(Λ , 𝒜 ) comprised of filtered locally trivial square-zero extensions of Λ by 𝒜 isdenoted by exal f (Λ , 𝒜 ) . Denote by(6) b 𝒞 • (Λ , 𝒜 ) := 𝒜 ⊗ C Λ ˆ ⊗• the bounded below Hochschild chain complex of the sheaf of algebras Λ with coefficients in the Λ − Λ -bimodule 𝒜 and by(7) b 𝒞 • (Λ , 𝒜 ) := ℋℴ𝓂 Λ e (Λ ˆ ⊗• +2 , 𝒜 ) ∼ = ℋℴ𝓂 C (Λ ˆ ⊗• , 𝒜 ) , where 𝒜 e := 𝒜 ⊗ C 𝒜 op , the bounded below complexes of continuous Hochschild cochains of Λ with values in 𝒜 ,respectively. Accordingly, we denote by ℋℋ • (Λ , 𝒜 ) and ℋℋ • (Λ , 𝒜 ) the homology sheaf of (6) and the cohomol-ogy sheaf of (7), respectively. If we view Λ ˆ ⊗• +2 as an acyclic bar resolution of Λ as a left Λ e -module, ℋℋ • (Λ , 𝒜 ) and ℋℋ • (Λ , 𝒜 ) can be expressed in terms of left and right derived functors ℋℋ • (Λ , 𝒜 ) = 𝒯ℴ𝓇 Λ e • ( 𝒜 , Λ) = H • ( 𝒜 ⊗ L Λ e Λ) (8) ℋℋ • (Λ , 𝒜 ) = ℰ𝓍𝓉 • Λ e (Λ , 𝒜 ) = H • ( R ℋℴ𝓂 Λ e (Λ , 𝒜 )) . (9)In case that Λ and 𝒜 are equipped with an increasing filtration, there is a natural increasing filtration on the n -foldtensor product Λ ⊗ n for every integer n ≥ by F p (Λ ⊗ n ) = ⊕ i + ··· i n = p F i Λ ⊗ · · · ⊗ F i n Λ . The restriction of the Hochschild cochains of Λ with values in 𝒜 to filtration preserving cochains yields a subcomplex b 𝒞 • f (Λ , 𝒜 ) of b 𝒞 • (Λ , 𝒜 ) which is equipped with a canonical decreasing filtration b 𝒞 • f (Λ , 𝒜 ) = F b 𝒞 • f (Λ , 𝒜 ) ⊃ F b 𝒞 • f (Λ , 𝒜 ) ⊇ · · · ⊇ F p b 𝒞 • f (Λ , 𝒜 ) ⊇ · · · given by F p b 𝒞 nf (Λ , 𝒜 ) := (cid:8) f ∈ ℋℴ𝓂 C (Λ ⊗ n , 𝒜 ) : f ( F q (Λ ⊗ n )) ⊆ F q − p 𝒜 for every q (cid:9) for every n ≥ . We denote the cohomology sheaf of b 𝒞 • f (Λ , 𝒜 ) by ℋℋ • f (Λ , 𝒜 ) . Theorem 2.10.
There is an isomorphism of C -vector spaces between exal f (Λ , 𝒜 ) and H ( X, σ ≥ b 𝒞 • f (Λ , 𝒜 )) .Proof. The proof is verbatim a repetition of the proof of the first half of Theorem . in [Gra61] adapted to the filteredcase. Nevertheless, for the sake of completeness, we shall provide a detailed proof.Let K •• := ˇC • (cid:0) 𝒰 , b 𝒞 • f (Λ , 𝒜 ) (cid:1) be the Čech double complex associated to 𝒰 : ... ... ... ˇC i +1 f ( 𝒰 , b 𝒞 f (Λ , 𝒜 )) · · · ˇC i +1 f (cid:0) 𝒰 , b 𝒞 jf (Λ , 𝒜 ) (cid:1) ˇC i +1 f (cid:0) 𝒰 , b 𝒞 j +1 f (Λ , 𝒜 ) (cid:1) · · · ˇC if ( 𝒰 , b 𝒞 f (Λ , 𝒜 )) · · · ˇ C if (cid:0) 𝒰 , b 𝒞 jf (Λ , 𝒜 ) (cid:1) ˇC if (cid:0) 𝒰 , b 𝒞 j +1 f (Λ , 𝒜 ) (cid:1) · · · ... ... ... δ d d δ d δ dδ d d δ d δ dδ δ δ n which δ is the Čech differential and d is the standard Hochschild differential with δd − dδ = 0 whose total complex Tot • ( K •• ) has a differential D ′ := δ p,q + ( − p d p,q in bidegree ( p, q ) and differential D ′ tot = P p + q = n δ p,q +( − p d p,q in total degree n . Let → 𝒜 i −→ Γ p −→ Λ → be an element in exal f (Λ , 𝒜 ) . Then by definition thereis an open cover 𝒰 = { U α } of X together with filtered morphisms of sheaves of vector spaces j α : Λ | U α → Γ | U α for every α . On intersection U αβ := U α ∩ U β , define the filtered map h αβ := j β − j α . As p ◦ h αβ = 0 , itfollows that Im( h αβ ) = Im( i ) ∼ = 𝒜 . Hence, abusing notation we obtain a morphism of sheaves of C -vector spaces h αβ : Λ | U αβ → 𝒜 | U αβ . Let δ denote the Čech differential in the Čech complex ˇC • ( 𝒰 , b 𝒞 f (Λ , 𝒜 )) . Then, δ , ( h ) αβγ = ( j γ − j β ) − ( j γ − j α ) + ( j β − j α ) = 0 (10)implies that h αβ is a Čech -cocycle in ˇC ( 𝒰 , b 𝒞 f (Λ , 𝒜 )) . Define a map f α : Λ | U α ⊗ C Λ | U α → 𝒜 | U α as thecomposition Λ | U α ⊗ C Λ | U α j α ⊗ C j α −−−−−→ Γ | U α ⊗ C Γ | U α mult −−−→ Γ | U α q α −→ 𝒜 | U α where mult denotes the product in Γ and q α := id Γ − j α ◦ p . The associativity of the product in Γ implies that(11) d , f α = 0 . As explained in the proof of [Gra61, Theorem 3.2], substituting in the definition of f α the identities j β = j α − h αβ , q β = q β + h αβ ◦ p and p ◦ mult = mult( p ⊗ p ) yields ( δ , f ) αβ = f β − f α = h βα − h αβ = d , h αβ which together with Equality (10) and Equality (11) implies D ′ tot ( f α ⊕ h αβ ) = ( δ , − d , ) h αβ + ( δ , + d , ) f α = 0 .This means that f α ⊕ h αβ is a -cocycle in Tot • ( F K •• ) . Suppose → 𝒜 i ′ −→ Γ ′ p ′ −→ Λ is an equivalent extensionsuch that the diagram Γ0 𝒜 Λ 0Γ ′ k pi i ′ p ′ commutes and with associated induced -cocycle h ′ αβ ⊕ f ′ α in Tot • ( F K •• ) . If we identify 𝒜 with the imagesof the identity inclusions i ( 𝒜 ) and i ′ ( 𝒜 ) then k | 𝒜 = id 𝒜 . Let j α and j ′ α be the corresponding filtered splittinghomomorphisms of the filtered square-zero extensions. Set t α := j ′ α − k ◦ j α . Then we have, p ′ ◦ t α = id Λ | U α − id Λ | U α = 0 which implies Im( t α ) ∼ = Im( i ′ ) ∼ = 𝒜 . Hence, abusing notation this map extends to a linear morphism t α : Λ | U α → 𝒜 | U α , i.e. t α ∈ ˇC ( 𝒰 , b 𝒞 f (Λ , 𝒜 )) . Let as before h αβ := j β − j α , and h ′ αβ := j ′ β − j ′ α . Note that k ◦ h αβ = h αβ .Then, we have, h ′ αβ − h αβ = h ′ αβ − k ◦ h αβ = ( j ′ β − k ◦ j β ) − ( j ′ α − k ◦ j α )= δ ( t α ) . (12)Consider the difference of both (0 , -cochains f α and f ′ α (cid:0) f ′ α − f α (cid:1) ( λ , λ ) = q ′ α ◦ mult ′ ( j ′ α ( λ ) ⊗ j ′ α ( λ )) − q α ◦ mult( j α ( λ ) ⊗ j α ( λ ))= (id Γ ′ − j ′ α ◦ p ′ )( j ′ α ( λ ) j ′ α ( λ )) − (id Γ − j α ◦ p )( j α ( λ ) j α ( λ ))= j ′ α ( λ ) j ′ α ( λ ) − j ′ α ( λ λ ) + j α ( λ λ ) − j α ( λ ) j α ( λ )= (cid:0) j ′ α ( λ ) j ′ α ( λ ) − j ′ α ( λ ) kj α ( λ ) (cid:1) + j ′ α ( λ ) kj α ( λ ) + (cid:0) j ′ α ( λ ) j α ( λ ) − kj α ( λ ) j α ( λ ) (cid:1) − j ′ α ( λ ) j α ( λ ) − j ′ α ( λ λ ) + kj α ( λ λ )= j ′ α ( λ ) t α ( λ ) + t α ( λ ) j α ( λ ) − t α ( λ λ )= λ · t α ( α ) − t α ( λ λ ) + t α ( λ ) · λ dt α ( λ , λ ) (13)where we used that k | 𝒜 = id 𝒜 and in the second to the last line we used that j α ( λ ) aj ′ α ( λ ) = λ · a · λ forall sections λ , λ of Λ and every section a of 𝒜 due to the fact that the extension is by definition square-zero.Identities (12) and (13) imply that both -cocycles h αβ ⊕ f α and h ′ αβ ⊕ f ′ α differ by a coboundary D ′ tot t α = ( δ + d ) t α in Tot • ( F K •• ) . Hence, there is a well-defined mapping(14) exal f (Λ , 𝒜 ) → ˇH ( 𝒰 , b 𝒞 • f (Λ , 𝒜 )) . Conversely, suppose f α ⊕ h αβ is a -cocycle in Tot • ( F K •• ) . Then we can define a locally trivial filtered square-zero C -vector space extension of Λ by 𝒜 as laid out in [Gra61, Proposition 3.1]. Let Γ be the sheaf defined by [ α (Λ ⊕ 𝒜 ) | U α (cid:30)(cid:28) ( λ, a ) | U α ∼ ( λ, a + h αβ ( λ )) | U α : ( λ, a ) | U α ∈ (Λ ⊕ 𝒜 ) | U α (cid:29) The product in Γ | U α is given by ( λ , a ) | U α · ( λ , a ) | U α = ( λ λ , λ a + a λ + f α ( λ , λ )) | U α . The well-definitionof the product on intersections U αβ follows from the compatibility relation for f α . With this multiplication theabove construction becomes in fact an algebra extension. Assume that h ′ αβ ⊕ f ′ α is a cohomologous -cocycle in Tot • ( F K •• ) . Then, there is by definition a (0 , -cochain t α such that h ′ αβ ⊕ f ′ α − h αβ ⊕ f α = D ′ tot t α . Consequently,we can write for the associated locally trivial filtered square-zero extension Γ ′ of Λ by 𝒜 [ α (Λ ⊕ 𝒜 ) | U α (cid:30)(cid:28) ( λ, a ) | U α ∼ ( λ, a + h αβ ( λ ) + δ ( t α )( λ )) | U α : ( λ, a ) | U α ∈ (Λ ⊕ 𝒜 ) | U α (cid:29) We define a morphism k : Γ → Γ ′ of sheaves of C -modules by ( λ, a ) | U α ( λ, a + t α ( λ )) | U α . This map is in fact amorphism of sheaves of C -algebras since we have k (cid:0) ( λ , a ) | U α ( λ , a ) | U α (cid:1) = k (cid:0) ( λ λ , λ a + a λ + f α ( λ , λ )) | U α (cid:1) = ( λ λ , λ a + a λ + f α ( λ , λ ) + t α ( λ λ )) | U α = ( λ λ , λ a + a λ + dt α ( λ , λ ) + t α ( λ λ ) + f ′ α ( λ , λ )) | U α = ( λ , a + t α ( a )) | U α · ( λ , a + t α ( a )) | U α = k (( λ , a ) | U α ) k (( λ , a ) | U α ) . As evidently k ◦ i = i ′ and p ′ ◦ k = p , this morphism defines an equivalence relation between the extensions Γ and Γ ′ . Thus, we obtain a well-defined map(15) ˇH ( 𝒰 , b 𝒞 • f (Λ , 𝒜 )) → exal f (Λ , 𝒜 ) . It is evident that the morphisms (14) and (15) are the inverses of each other. As direct limits preserve exactness, weconclude exal f (Λ , 𝒜 ) ∼ = ˇH ( X, b 𝒞 • f (Λ , 𝒜 )) . The claim follows from the fact that X is paracompact by assumption. (cid:3) In the spacial case of 𝒜 = Λ , inspired by the isomorphism in Lemma 2.5, we define the space of first-order defor-mations 𝒟ℯ𝒻 (Λ) of Λ as exal(Λ , Λ) and the space of filtered first-order deformations 𝒟ℯ𝒻 f (Λ) of Λ by exal f (Λ , Λ) .
3. Universal deformation of 𝒟 X ⋊ G Sheaves of Calabi-Yau algebras.
Calabi-Yau algebras were introduced and first studied in [Gin06]. Thesealgebras arise naturally in the theory of non-commutative deformation of spaces and possess a number of valaubleproperties. In this note we are interested in them because of the duality between Hochschild homology and coho-mology of Calabi-Yau algebras which enables us to derive a cohomological holomorphic version of Engeli-Felder’strace density morphism. Let us recall the definition of a Calabi-Yau algebra from [Gin06].
Definition 3.1 (Definition 3.2.3, [Gin06]).
An associative k -algebra is a Calabi-Yau algbra of dimension d if Λ hasa finitely-generated projective Λ − Λ -bimodule resolution and HH • (Λ , Λ ⊗ k Λ) ∼ = Λ[ − d ] as a graded Λ − Λ -bimodule. As stated earlier Calabi-Yau algebras admit a so-called van den Berg duality between Hochschild homology andcohomology which is precisely formulated in the ensuing theorem. heorem 3.2 ([Van98]). Let Λ be a Calabi-Yau associative C -algebra of dimension d . Then for every left Λ e -module M there is a canonical isomorphism HH i (Λ , M ) ∼ = HH d − i (Λ , M ) . The above concepts admits a natural generalization to sheaves of associative C -algebras. Definition 3.3.
A sheaf of associative C -algebras Λ on a complex space Y is Calabi-Yau of dimension d provided thatit admits a locally free left Λ e resolution and ℰ𝓍𝓉 • Λ e (Λ , Λ ⊗ C Λ) ∼ = Λ[ − d ] . Let 𝒫 • → Λ denote the finitely-generated projective left Λ e resolution of Λ . It is evident that for every y ∈ Y the stalk Λ y is a Calabi-Yau algebra in the sense of Definition 3.1. Indeed, we have HH • (Λ y , Λ y ⊗ C y Λ y ) = Ext Λ ey (Λ y , Λ y ⊗ C y Λ y )= H • ( ℋℴ𝓂 Λ e ( 𝒫 • , Λ ⊗ C Λ)) y = ℰ𝓍𝓉 • Λ e (Λ , Λ ⊗ C Λ) y = Λ y [ − d ] . This taken together with Theorem 3.2 in turn deliever the following immediate result.
Lemma 3.4.
Let Λ be a Calabi-Yau sheaf of associative algebras of dimension d on a complex space Y and let 𝒜 be a left Λ e -module. Then ℋℋ i (Λ , 𝒜 ) ∼ = ℋℋ d − i (Λ , 𝒜 ) . Later we shall need the following fact about the 𝒪 X -algebra Sym • ( 𝒯 X ) . Proposition 3.5.
The sheaf of 𝒪 X -algebras Sym • ( 𝒯 X ) is a sheaf of Calabi-Yau C -algebras of dimension n .Proof. We start by constructing a locally free resolution of
Sym • ( 𝒯 X ) of length n in terms of left Sym • ( 𝒯 X ) e -modules. Put Y = X × X with the projection p : Y → X on the first term and p : Y → X on the second term of Y . Let δ : X → Y, x ( x, x ) be the diagonal embedding with a diagonal ∆ := Im( δ ) . Since X is a complex analyticmanifold, ∆ is closed and hence analytic in Y . It, therefore, defines a sheaf of ideals ℐ ∆ ⊂ 𝒪 Y with a zero set ∆ . Let W be an open set in the product topology of Y and let f , . . . , f n , g , . . . , g n ∈ ℐ ∆ | W . Let ( x , . . . , x n , y , . . . , y n ) be local coordinates on W . Set a local section s | W := P ni =1 f i ∂∂x i + P g j ∂∂y j of the tangent sheaf 𝒯 Y . It induces a 𝒪 Y | W -module homomorphism s ∗ : 𝒯 ∗ Y | W → 𝒪 Y | W ω s ∗ ( ω ) := h ω, s i . where h· , ·i is the pairing of 1-forms and vector fields. From the definition of s ∗ it is evident that Im( s ∗ ) = ℐ ∆ whichidentifies ∆ with the zero submanifold of s . By the standard theory of Koszul resolutions of zero submanifolds ( cf. [ FT89 ]) we get a resolution n ^ 𝒯 ∗ Y → · · · → ^ 𝒯 ∗ Y → 𝒪 Y → 𝒪 Y / ℐ ∆ (16)of 𝒪 Y / ℐ ∆ of length n on Y . Denote the external tensor product p ∗ Sym • 𝒯 X ⊗ 𝒪 Y p ∗ Sym • 𝒯 X of Sym • 𝒯 X by Sym • 𝒯 X ⊠ Sym • 𝒯 X . It is per definition a locally free 𝒪 Y -module. Hence, it is a flat module over 𝒪 Y . Ergo,tensoring the exact sequence (16) with Sym • 𝒯 X over 𝒪 Y yields an exact sequence of 𝒪 Y -modules Sym • 𝒯 X ⊠ Sym • 𝒯 X ⊗ 𝒪 Y n ^ 𝒯 ∗ Y → . . . → Sym • 𝒯 X ⊠ Sym • 𝒯 X ⊗ 𝒪 Y ^ 𝒯 ∗ Y (17) → Sym • 𝒯 X ⊠ Sym • 𝒯 X → Sym • 𝒯 X ⊠ Sym • 𝒯 X ⊗ 𝒪 Y ⊗ 𝒪 Y / ℐ ∆ (18)Let j ∆ : ∆ ֒ → Y be the closed embedding and let ∗| ∆ denote the corresponding restriction of sheaves to the closedanalytic submanifold ∆ . As the inverse image of j ∆ is an exact functor, we get an exact sequence of 𝒪 Y | ∆ -modules j − (Sym • 𝒯 X ⊠ Sym • 𝒯 X ) ⊗ 𝒪 Y | ∆ n ^ 𝒯 ∗ Y | ∆ → · · · → ^ j − (cid:0) Sym • 𝒯 X ⊠ Sym • 𝒯 X (cid:1) ⊗ 𝒪 Y | ∆ 𝒯 ∗ Y | ∆ A complex analytic manifold and an orbifold are special examples of a complex space. j − (cid:0) Sym • 𝒯 X ⊠ Sym • 𝒯 X (cid:1) → j − (cid:0) Sym • 𝒯 X ⊠ Sym • 𝒯 X (cid:1) ⊗ 𝒪 Y | ∆ ⊗ 𝒪 ∆ (19)where 𝒪 ∆ := j − (cid:0) 𝒪 Y / ℐ ∆ (cid:1) . The last term in Sequence (19) can be rewritten as j − (cid:0) Sym • 𝒯 X ⊠ Sym • 𝒯 X (cid:1) ⊗ 𝒪 Y | ∆ 𝒪 ∆ == 𝒪 Y | ∆ ⊗ ( p j ∆ ) − 𝒪 X ⊗ C ( p j ∆ ) − 𝒪 X ( p j ∆ ) − Sym • ( 𝒯 X ) ⊗ C ( p j ∆ ) − Sym • ( 𝒯 X ) ⊗ 𝒪 Y | ∆ 𝒪 ∆ ∼ = 𝒪 Y | ∆ ⊗ ( p j ∆ ) − 𝒪 X ⊗ C ( p j ∆ ) − 𝒪 X Sym • ( j − ( p − 𝒯 X ⊕ p − 𝒯 X )) ⊗ 𝒪 Y | ∆ 𝒪 ∆ ∼ = Sym • ( 𝒯 Y | ∆ ) ⊗ 𝒪 Y | ∆ 𝒪 ∆ ∼ = Sym • ( 𝒯 ∆ ) (20)where 𝒯 ∆ is the tangent sheaf on the diagonal submanifold ∆ . As an isomorphism of abelian categories between 𝒪 ∆ − Mod and 𝒪 X − Mod the 𝒪 ∆ -module pullback δ ∗ is exact, too. This way, applying δ ∗ on (19) and plugging (20)in (19), we obtain an exact sequence of 𝒪 X -modules δ ∗ (cid:16) j − (Sym • 𝒯 X ⊠ Sym • 𝒯 X ) ⊗ 𝒪 Y | ∆ • ^ 𝒯 ∗ Y | ∆ (cid:17) → δ ∗ (cid:16) Sym • ( 𝒯 ∆ ) (cid:17) . (21)which accounting for δ ∗ (cid:16) j − (Sym • 𝒯 X ⊠ Sym • 𝒯 X ) ∼ = Sym • ( 𝒯 X ) ⊗ C Sym • ( 𝒯 X ) becomes the same as Sym • ( 𝒯 X ) ⊗ C Sym • ( 𝒯 X ) ⊗ δ − 𝒪 Y | ∆ • ^ δ − 𝒯 ∗ Y | ∆ → Sym • ( 𝒯 X ) . (22)The exact sequences of locally free 𝒪 X -modules (21), respectively (22) can be seen as the desired Koszul type resolu-tion of Sym • ( 𝒯 X ) of length n in terms of locally free left Sym • ( 𝒯 X ) e -modules. The Calabi-Yau property followsconsequently from ℰ𝓍𝓉 • Sym • ( 𝒯 X ) e (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) ⊗ C Sym • ( 𝒯 X )) :== H • (cid:0) R ℋℴ𝓂
Sym • ( 𝒯 X ) e (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) ⊗ C Sym • ( 𝒯 X ) (cid:1) ∼ = H • (cid:16) Sym • ( 𝒯 X ) ⊗ C Sym • ( 𝒯 X ) ⊗ δ − 𝒪 Y | ∆ • ^ δ − 𝒯 ∗ Y | ∆ (cid:17) = Sym • ( 𝒯 X )[ − n ] (23)which completes the proof of the statement in the proposition. (cid:3) With the help of Proposition (3.5) we show that the sheaf of holomorphic differential operators 𝒟 X on a complexanalytic manifold X is Calabi-Yau of dimension n . Proposition 3.6. 𝒟 X is a sheaf of Calabi-Yau algebras of dimension n .Proof. Since 𝒟 X is a filtered quantization of Sym • ( 𝒯 X ) , the exact sequence (21) from Proposition 3.5 can be rewrittenin the form δ ∗ (cid:16) Gr • (cid:0) j − ( 𝒟 X ⊠ 𝒟 X ) (cid:1) ⊗ 𝒪 Y | ∆ • ^ 𝒯 ∗ Y | ∆ (cid:17) → δ ∗ (cid:16) Gr • ( 𝒟 ∆ ) (cid:17) . (24)It is a known result from homological algebra that for any filtered complex C • , exactness of the associated gradedof C • implies exactness of C • . Hence, δ ∗ (cid:16) j − ( 𝒟 X ⊠ 𝒟 X ) ⊗ 𝒪 Y | ∆ • ^ 𝒯 ∗ Y | ∆ (cid:17) → δ ∗ (cid:16) 𝒟 ∆ (cid:17) . (25)is an exact sequence of locally free 𝒪 X -modules. In particular, this acyclic complex is the wanted Koszul type resolu-tion of length n of 𝒟 X in terms of locally free left 𝒟 eX -modules. The Calabi-Yau property follows in an analogousmanner to Proposition 3.5. (cid:3) An immediate consequence of Proposition 3.6 and Proposition 3.5 is the following result.
Corollary 3.7.
The sheaves ( 𝒟 X ) G and 𝒟 X ⋊ G as well as their corresponding associated graded Sym • 𝒪 X ( 𝒯 X ) G and Sym • 𝒪 X ( 𝒯 X ) ⋊ G on X/G are sheaves of Calabi-Yau C -algebras of dimension C X . roof. Taking the invariants with respect to the action of G is an exact functor ( · ) G : Bimod ( 𝒟 X ) → Bimod ( 𝒟 GX ) .Hence, applying ( · ) G on (25) yields a locally free resolution of ( 𝒟 X ) G on X/G in terms of ( 𝒟 X ) G − ( 𝒟 X ) G -bimodules of length n . With the proper locally free left 𝒟 eX -module resolution of 𝒟 X at hand we obtain ℰ𝓍𝓉 • ( 𝒟 X ) Ge (( 𝒟 X ) G , ( 𝒟 X ) G ⊗ C ( 𝒟 X ) G ) = ( 𝒟 X ) G [ − n ] which implies that ( 𝒟 X ) G is Calabi-Yau of dimension n . The Morita equivalence between ( 𝒟 X ) G and 𝒟 X ⋊ G implies the latter statement. The proof of the claim for the associated graded is literally identical. (cid:3) Corollary 3.8.
Let Θ be a Hochschild chain n -cycle of 𝒟 X ⋊ G . Then the morphism of cochain complexes µ : 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) → 𝒞 n −• ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) f Θ ∩ f is a quasi-isomorphism, where 𝒞 n −• ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) is viewed as cochain complex by inverting degrees.Proof. It follows from the Calabi-Yau propert of 𝒟 X ⋊ G together with the fact that µ ∗ is injective and the cohomologygroups are finite-dimensional. (cid:3) Trace density morphisms.
Generalized trace density morphism for 𝒟 X . Let in the following ( 𝒜 • X, C , d dR ) denote the de Rham complex ofsheaves of smooth complex-valued differential forms with differential d dR , and (Ω • X , d dR ) denote the complex ofsheaves of holomorphic differential forms on X with differential d dR .We start by adapting Engeli-Felder’s trace density map constructed in [EF08] to the holomorphic setting. In theholomorphic setting the term trace density is not justified because then the images of these maps are hypercoho-mology classes which cannot easily be integrated like de Rham cohomology classes in the complex valued smoothcase. Therefore, we use the vague term generalized trace density morphism to describe maps from the Hochschild(co)chain complex of 𝒟 X to the complex of sheaves of holomorphic de Rham differential forms on T ∗ X . We shallcontinue using the term generalized trace density morphism when dealing later in the next subsection with similarmaps for 𝒟 X ⋊ G .Let Ω be the Aut n -invariant W n -valued holomorphic connection -form on the Harish-Chandra ( W n , Aut n ) -torsor π coor : X coor → X of formal coordinates on X as defined in [BR73]. This induces a flat holomorphic connection d + [Ω , · ] on X coor × b 𝒟 n . By means of Gelfand-Kazhdan formal geometry one shows that there is an isomorphismof sheaves between 𝒟 X and the sheaf π coor ∗ 𝒪 flat ( X coor × b 𝒟 n ) of flat sections of the algebra bundle X coor × b 𝒟 n . Forthe purposes of a definition of a generalized trace density map it is more convenient to express 𝒟 X at least locallyin terms of flat sections of a bundle over the cotangent bundle T ∗ X and not over X coor . We do that in the followingway. The definition of X coor as an inverse limit of a projective system of of GL n -equivariant holomorphic sub-mersions X k → X k − yields a GL n ( C ) -equivariant holomorphic map X coor → Fr( X ) where Fr( X ) is the framebundle of X and corresponds to X in the inverse system. Let 𝒰 = { U α } be an open cover of X which trivializesthe tangent bundle, hence, the cotangent bundle π : T ∗ X → X and the frame bundle Fr( X ) → X , too. On a smallenough U α ∈ 𝒰 there is always a local GL n ( C ) -equivariant holomorphic section s α : Fr( X ) | U α → X coor | U α . Thepullback of Ω along s α yields a GL n ( C ) -equivariant holomorphic -form ω α := s ∗ α Ω on Fr( X ) | U α with values in W n which satisfies the Maurer-Cartan equation. It descends to a holomorphic W n -valued Maurer-Cartan -form on U α , which for brevity we also denote by ω α , and induces a holomorphic flat connection ∇ α on the associated bundle D α := Fr( X ) | U α × GL n ( C ) b 𝒟 n → U α . We take finally the pullback holomorphic bundle π ∗ D α → T ∗ X | U α which isflat with respect to π ∗ ∇ α . The direct image under π of the sheaf of pullback horizontal sections of π ∗ D α with respectto π ∗ ∇ α is isomorphic to π coor ∗ 𝒪 flat ( X coor × b 𝒟 n ) | U α and hence its sections are in one-to-one correspondence withholomrphic differential operators in 𝒟 X ( U α ) . The pullback of ω α ∈ Ω X ( U α ) along the holomorphic projection π induces a holomorphic -form on T ∗ X | U α satisfying the Maurer-Cartan equation which again by abuse of notationwe denote by ω α .Adhering to the notation in [EF08] we shall denote k -chains of the normalized Hochschild chain complex of anassociative algebra A by ( a , a , · · · , a k ) . Recall the nontrivial Feigin-Felder-Schoikhet normalized Hochschild n -cocycle τ n of 𝒟 n with values in the dual bimodule 𝒟 ∗ n which naturally extends to a linear form on b C n ( 𝒟 n ) := 𝒟 ˆ ⊗ n +1 n . It is GL n ( C ) -basic meaning that it is invariant under the adjoint action of GL n ( C ) on 𝒟 n and for any ∈ gl n ( C ) , n X j =1 ( − j τ n ( b D , · · · , b D j − , a, b D j , · · · , b D n − ) = 0 , where b D j ∈ 𝒟 n for all j = 0 , . . . , n − , holds true.For every intersection U αβ = U α ∩ U β , let g αβ : U αβ → GL n ( C ) be the corresponding transition map between thefiber coordinates of T ∗ X . The relation between two local -forms ω α and ω β defined in coordinates of T ∗ X | U α and T ∗ X | U β is given by ω β = Ad( g − αβ ) ω α + g − αβ dg αβ . Furthermore, every flat section b D α of D α over U α representing a unique differential operator D α in 𝒟 X ( U α ) trans-forms under a change of trivialization according to b D β = Ad( g − αβ ) b D α . Now, consider on every open set U α the following composition of morphisms C p ( 𝒟 X ( U α )) ֒ → C p (Γ flat ( π − ( U α ) , π ∗ D α )) ֒ → C p (cid:0) (Ω • T ∗ X ( π − ( U α ) , π ∗ D α ) , ˜ ∇ α ) (cid:1) → C p (cid:0) (Ω • T ∗ X ( π − ( U α ) , b 𝒟 n ) , d dR + [ ω α , · ]) (cid:1) → C p (cid:0) (Ω • T ∗ X ( π − ( U α ) , b 𝒟 n ) , d dR ) (cid:1) → (Ω n − pT ∗ X ( π − ( U α )) , ( − n − p d dR ) → (Ω n − pT ∗ X ( π − ( U α )) , d dR ) , (26)where ˜ ∇ α is the covariant derivative induced by ∇ α , with corresponding connecting morphisms given by ( D , . . . , D p ) b · ( b D , . . . , b D p ) id ( b D , . . . , b D p ) id ( b D , . . . , b D p ) X k ≥ ( − k ( b D , . . . , b D p ) × ( ω α ) k τ n (cid:0) X k ≥ ( − k ( b D , . . . , b D p ) × ( ω α ) k (cid:1) ( − ⌊ n − p ⌋ X k ≥ ( − ⌊ k +12 ⌋ τ n (cid:0) ( b D , . . . , b D p ) × ( ω α ) k (cid:1) . Each of the above morphism are discussed in a greater detail in [Ram11, Section 3] This composition gives a mapbetween complexes of sheaves over U α χ α : b 𝒞 • ( 𝒟 X , 𝒟 X ) | U α −→ π ∗ Ω n −• T ∗ X | Uα ( D , · · · , D n ) X k ≥ ( − ⌊ k +12 ⌋ τ n (cid:0) ( b D , · · · , b D n ) × ( ω α ) k (cid:1) where ( ω α ) k denotes the normalized Hochschild k -chain (1 , ω α , · · · , ω α ) ( k copies of ω α ). We now want to showthat the map χ α extends to a map χ over all of X . To that aim, consider the following easy to prove binomial formulafor the shuffle product ( a + b ) k = k X j =0 (cid:18) kj (cid:19) ( a ) k − j × ( b ) j which we invoke in the ensuing computation. After a change of trivialization χ α changes as χ β ( D , · · · , D p ) = X k ≥ ( − ⌊ k +12 ⌋ τ n (cid:16) ( b D β , · · · , b D pβ ) × ( ω β ) k (cid:17) = X k ≥ ( − ⌊ k +12 ⌋ τ n (cid:16) (Ad( g − αβ ) b D α , · · · , Ad( g − αβ ) b D pα ) × (Ad( g − αβ ) ω α − g − αβ dg αβ ) k (cid:17) = ( − ⌊ n + − p ⌋ τ n (cid:16) (Ad( g − αβ ) b D α , · · · , Ad( g − αβ ) b D pα ) × (Ad( g − αβ ) ω α − g − αβ dg αβ ) n − p (cid:17) = ( − ⌊ n + − p ⌋ τ n (cid:16) (Ad( g − αβ ) b D α , · · · , Ad( g − αβ ) b D pα ) × n − p X j =0 (cid:18) n − pj (cid:19) ( − j (Ad( g − αβ ) ω α ) n − p − j × ( g − αβ dg αβ ) j (cid:17) ( − ⌊ n + − p ⌋ τ n (cid:16) (Ad( g − αβ ) b D α , · · · , Ad( g − αβ ) b D pα ) × (Ad( g − αβ ) ω α ) n − p (cid:17) ++ n − p X j =1 ( − j + ⌊ n + − p ⌋ (cid:18) n − pj (cid:19) τ n (cid:16) (Ad( g − αβ ) b D α , · · · , Ad( g − αβ ) b D pα ) × (Ad( g − αβ ) ω α ) n − p − j × ( g − αβ dg αβ ) j − × ( g − αβ dg αβ ) (cid:17) = ( − ⌊ n + − p ⌋ τ n (cid:16) ( b D α , · · · , b D pα ) × ( ω α ) n − p (cid:17) = χ α ( D , · · · , D p ) (27)where in the second to the last line we made use of the fact that the Feigin-Felder-Schoikhet n -cocycle τ n is GL n ( C ) -basic. This demonstrates that the map is independent on the choice of trivialization and hence well definedas a morphism of complexes of sheaves on { U α } . Consequently, the family of maps ( U α , χ α ) extends to a morphism(28) χ : b 𝒞 • ( 𝒟 X ) → π ∗ Ω n −• T ∗ X of cochain complexes of sheaves on the whole of X if we consider the Hochschild chain complex b 𝒞 • ( 𝒟 X ) as acochain complex by inverting degrees, i.e. b 𝒞 −• ( 𝒟 X , 𝒟 X ) = b 𝒞 • ( 𝒟 X , 𝒟 X ) . For any choice of local holomorphicsections { s α } on the different members of 𝒰 the corresponding induced cochain morphisms (28) are homotopic.Therefore, at the level of cohomology sheaves the morphism χ ∗ is canonic. Moreover, we can prove the ensuingresult. Proposition 3.9.
The morphism (28) is a quasi-isomorphism.Proof.
The stalk of the cohomology sheaf H • ( π ∗ Ω n −• T ∗ X ) at x ∈ X by definition is H −• ( π ∗ Ω n −• T ∗ X ) x ∼ = H −• ( π ∗ Ω n −• T ∗ X,π ( x,v ) ) ∼ = lim −→ π − ( U ) ∋ ( x,v ) H −• (Ω n −• T ∗ X ( π − ( U ))) ∼ = lim −→ X ⊇ U ∋ x H −• (Ω n −• X ( U )) ∼ = H −• (Ω n −• X,x ) ∼ = H −• ( 𝒜 n −• X, C ,x )= ( C , if • = 2 n , otherwisewhere in the second and the fourth line we use the commutativity of direct limits with the cohomology functor, inthe third line we invoke the homotopy equivalence of the holomorphic cotangent bundle and the base space and inthe second to the last line we invoke the fact that 𝒜 • X, C and Ω • X are both (injective) resolutions of the constant sheaf C X . Consider the induced map on the cohomology sheaves χ ∗ : ℋℋ −• ( 𝒟 X , 𝒟 X ) → H −• ( π ∗ Ω n −• T ∗ X ) . This map is an isomorphism if and only if it is an isomorphism on stalks. Let ( x , . . . , x n ) be the local coordinatesat a point x ∈ X . From Theorem in [Wod87] it follows that for x ∈ X , ℋℋ −• ( 𝒟 X , 𝒟 X ) x ∼ = H −• ( b 𝒞 −• ( 𝒟 X,x , 𝒟 X,x )) ∼ = HH • ( 𝒟 X,x ) ∼ = lim −→ X ⊇ U ∋ x HH • ( 𝒟 X ( U )) ∼ = ( lim −→ X ⊇ U ∋ x C · { c X ( U ) } , if • = 2 n , otherwise ∼ = ( C · { c X,x } , if • = 2 n , otherwise(29) here c X ( U ) := P π ∈ S n sgn( π )(1 , u π (1) , . . . , u π (2 n ) ) with u j − = ∂ x j , u j = x j is the generator of HH n ( D X ( U )) .By the normalization property of τ n , for every x ∈ X , we have on the generator c X,x of HH n ( 𝒟 X,x ) χ ∗ ,x ([ c X,x ]) = [1] which implies that the induced morphism χ ∗ ,x : ℋℋ −• ( 𝒟 X , 𝒟 X ) x → H −• ( π ∗ Ω n −• T ∗ X ) x is a non-trivial map between one-dimensional stalks of cohomology sheaves. Hence, χ ∗ ,x is an isomorphism. Ergo, χ ∗ is an isomorphism of cohomology sheaves which concludes the proof. (cid:3) Generalized trace density morphism for the skew group ring 𝒟 X ⋊ G . Sheaves on the global quotient
X/G such as 𝒟 X ⋊ G can equivalently be viewed as sheaves defined in the G -equivariant topology of X . To sim-plify computations, we shall equivalently work on a basis for the G -equivariant topology of X comprised of sets ind GH U α := G × H U α , where H is a subgroups of G arising as a pointwise stabilizer in X and { U α } is a familyof open H -invariant slices of X . We note that every H -invariant slice is in particular a K -invariant slice for everysubgroup K of H . We are especially interested in strata associated to well-generated complex reflection subgroupsand cyclic subgroups because, as shown in [Vit20, Corollary 1.5], the dimension of the group of linear traces ofthe (completed) Cherednik algebra associated to such groups is nontrivial which implies the existence of nontrivialtraces of the (formally completed) Cherednik algebra. These linear traces ultimately are one of the building blocksof the generalized trace density morphisms for 𝒟 X ⋊ G as we will see below.We now set the notation we shall employ untill the end of the paper. For any subgroup H of G , let X H be theisotropy type of H , consisting of points x in X with stabilizers isomorphic to H , and let X H denote the fixed pointsubmanifold of points in X whose stabilizers contain H . The connected components X iH of X H are referred to asstrata. Every X iH is open and dense in some connected component of X H which for brevity shall be designated alsoby X Hi . Let h g i be the cyclic group generated by the single element g in G and let X i h g i be a stratum of codimension l ig lying in the corresponding fixed point connected component X gi . Let π gi : T ∗ X gi → X gi be the cotangent bundleto X gi and let j gi : T ∗ X gi ֒ → T ∗ X , j gi : X gi ֒ → X be the respective canonical inclusions. We denote by j i h g i theobvious restrictions to the stratum X i h g i . Let N be the rank l ig normal bundle to X gi with structure group of itsframe bundle the centralizer Z of the lift of g in GL l ig ( C ) and let π coor : 𝒩 coor → X gi denote the coordinate bundleconsisting of infinite jets of parametrizations of N .Given a slice U α intersecting X gi , let ( x , . . . , x n − l ig ) denote the local coordinates along U α ∩ X gi and let ( y , . . . , y l ig ) be the local coordinates on U α in longitudinal direction to X gi . Let us further employ the notation 𝒜 h g i n − l ig ,l ig to signifythe Harish-Chandra ( W n − l ig ⋉ z ⊗ C [[ x , . . . , x n − l ig ]] , GL n − l ig ( C ) × Z ) -module b 𝒟 n − l ig ⊗ b 𝒟 l ig ⋊ h g i where W n − l ig isthe space of formal vector fields on a formal neighborhood of in C n − l ig and b 𝒟 n − l ig and b 𝒟 l ig are the rings of differen-tial operators on formal neighborhoods of in C n − l ig and C l ig , respectively. Let 𝒪 flat ( 𝒩 coor × 𝒜 h g i n − l ig ,l ig ) be the sheafof flat sections of the trivial bundle 𝒩 coor × 𝒜 h g i n − l ig ,l ig over 𝒩 coor with respect to the holomorphic flat connection d + [ ω, · ] , where ω is the holomorphic connection -form with values in the module 𝒜 h g i n − l ig ,l ig constructed in [Vit19,Section 4, Section 5] trivially extended to 𝒩 coor → X gi . It is a direct consequence of [Vit19, Theorem 6.3] that thereis a projection morphism p : 𝒟 X ⋊ G → j i h g i∗ π coor ∗ 𝒪 flat ( 𝒩 coor | X i h g i × 𝒜 h g i n − l ig ,l ig ) (30)Let us give an explicit description of the morphism on the basis of the G -equivariant topology. For a h g i -invariantslice U α at a point x on the stratum X i h g i , the morphism p assigns every element Dg of 𝒟 X (ind G h g i U α ) ⋊ G the uniqueflat section ˆ s of j i h g i∗ π coor ∗ 𝒪 flat ( 𝒩 coor | X i h g i × 𝒜 h g i n − l ig ,l ig ) over U α ∩ X i h g i from [Vit19, Theorem 6.3]. The definition of themorphism (30) in the case of a K -invariant slice U α at a point x on a stratum X jK which is contained in X i h g i ⊆ X gi is more subtle. As explained in [Vit19, Proposition 5.4], X jK is analytic within U α , ergo U α \ X jK is open in X . Recallthat by Cartan’s Lemma apart from X jK the slice U α intersects only strata associated to subgroups L of K . This factcombined with [Vit19, Corollary B.9] yields for every slice U β contained in U α \ X jK the following composition of lgebra morphisms 𝒟 X (ind GK U α ) ⋊ G ∼ = −→ C G ⊗ C K 𝒟 X ( U α ) ⋊ K ⊗ C K C G ։ 𝒟 X ( U α ) ⋊ K → 𝒟 X ( U α \ X jK ) ⋊ K ∼ = lim ←− U β iscentered at X L 𝒟 X (ind KL U β ) ⋊ K ։ Y U β is centeredat U α ∩ X i h g i { ( s ) ∈ 𝒟 X ( U β ) ⋊ h g i : res U β U γ ( s ) = s ′ for U β ⊆ U γ }→ j i h g i∗ π coor ∗ 𝒪 flat ( 𝒩 coor | X i h g i × 𝒜 h g i n − l ig ,l ig )( U α ∩ X i h g i ) . (31)The surjectivity of the fifth arrow follows from the definition of a projective limit and the fact that L -invariant slices U β with L not a subgroup of h g i , do not contain h g i -invariant slices. The last arrow in (31) is described as follows. Bythe gluing conditions in [Vit19, Section 6.2, Section 6.2], each element in 𝒟 X ( U β ) ⋊ h g i is uniquely represented by apair ( t | U β \ X i h g i , s | U β ∩ X i h g i ) in 𝒟 X ( U β \ X i h g i ) ⋊ h g i and π coor ∗ 𝒪 flat ( 𝒩 coor | X i h g i × 𝒜 h g i n − l ig ,l ig )( U β ∩ X i h g i ) , respectively.For all β , the collection of sections { s | U β ∩ X i h g i } coincide on every open set U γ ∩ X i h g i , contained in U β ∩ U β ′ ∩ X i h g i .Because of that, by the uniqueness axiom of sheaves the collection of sections s | U β ∩ X i h g i coincide on the whole ofevery intersection U β ∩ U β ′ ∩ X i h g i and the gluing axiom of sheaves imply the existence of a unique section s | U α ∩ X i h g i in π coor ∗ 𝒪 flat ( 𝒩 coor | X i h g i × 𝒜 h g i n − l ig ,l ig )( U α ∩ X i h g i ) . For the purposes of our work we need to extend the morphism (30)to the whole of X gi . Lemma 3.10.
The map (30) uniquely extends to a morpism of sheaves ¯ p : 𝒟 X ⋊ G → j gi ∗ π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 h g i n − l ig ,l ig ) . Proof.
We distinguish two cases: ) the codimension of the stratum X i h g i is equal to or bigger than , ) the stratum X i h g i is the prinicipal (dense and open) stratum ˚ X in X . ) Let U α be K -invariant linear slice centered on a stratum X jK contained in X gi as above. By Hartog’s Theoremthe third arrow in (31) is bijective. Hence, the images of the morphisms (31) and 𝒟 X ( U α \ X jK ) ⋊ h g i ∼ = −→ lim ←− U β iscentered at X L ,L< h g i 𝒟 X (ind h g i L U β ) ⋊ h g i ։ Y U β is centeredat U α ∩ X i h g i { ( s ) ∈ 𝒟 X ( U β ) ⋊ h g i : res U β U β ′ ( s ) = s ′ } → π coor ∗ 𝒪 flat ( 𝒩 coor | X i h g i × 𝒜 h g i n − l ig ,l ig )( U α ∩ X i h g i ) coincide. Thus, the preimage ˆ s in 𝒟 X ( W x \ X jK ) × h g i of every section ˆ s in the image of (31) is non-empty.Furthermore, as the codimension of X jK is at least in X , it follows 𝒟 X ( W x \ X jK ) ⋊ h g i ∼ = 𝒟 X ( W x ) ⋊ h g i by Hartog’sTheorem. By [Vit19, Theorem 6.3] ˆ s determines a unique section ˆ s of j Hi ∗ π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 h g i n − l ig ,l ig )( W x ∩ X gi ) with ˆ s | W x \ X jK = ˆ s . By the identity theorem, all sections ˆ s coincide on the open subset U α ∩ X i h g i of U α ∩ X gi ,ergo, on U α ∩ X gi . This gives a well-defined extension ¯ p . ) Assume X i h g i = ˚ X . Assume codim( X jK ) = 1 . Here, we only consider the case of a basic open sets ind GK U α with U α centered on X jK . Each element D c k in H ,c ( U α , K ) is uniquely represented by a pair of sections ˆ u c =( t | U α \ X jK k, s c | U α ∩ X jK ) of π coor ∗ 𝒪 flat ( ˚ X coor × b 𝒟 n )( U α \ X jK ) ⋊ K and π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 Kn − , )( U α ∩ X jK ) satisfyingthe gluing conditions from [Vit19, Section 6.1]. Setting c ( Y, s ) = 0 for all complex reflections s in K , the pair ˆ u describes a unique element in 𝒟 X ( U α ) ⋊ K . If we set Y = ˚ X ` X jK , the same element D k uniquelly determines flatsection ˆ rk in π coor ∗ 𝒪 flat ( Y coor × 𝒟 n )( U α ) ⋊ K . Clearly, the restriction maps satisfy res U α U α \ X jK (ˆ rk ) = res U α U α \ X jK (ˆ u ) = t | U α \ X jK k . Hence, ˆ r is the unique flat section in π coor ∗ 𝒪 flat ( Y coor × 𝒟 n )( U α ) which is assigned to D k and extends t | U α \ X jK . This gives the wanted extension ¯ p . We further extend (30) step by step to the union of ˚ X with all strata ofcodimension . Extensions beyond codimension follow from Hartog’s theorem. (cid:3) et 𝒰 = { U α } be a family of slices at points x on X gi with stabilizers H ⊇ h g i , which trivialize T X | X gi , and thesets W α = U α ∩ X gi form an open cover of X gi . We pullback ω along a locally holomorphic section s α of 𝒩 coor over W α = U α ∩ X gi to a 𝒜 h g i n − l ig ,l ig -valued holomorphic -form ω α := s ∗ α ω on W α satisfying the Maurer-Cartan equationin a similar fashion to Section 3.2.1. A further pullback of ω α along π gi yields a holomorphic Maurer-Cartan formon T ∗ X gi | W α with values in 𝒜 h g i n − l ig ,l ig , as desired. For each linearly independent trace φ of b 𝒟 l ig ⋊ h g i we define a GL n − l ig ( C ) × Z -basic normalized n − l ig Hochschild cocycle ψ n − l ig of 𝒜 h g i n − l ig ,l ig with values in the dual bimodule 𝒜 h g i∗ n − l ig ,l ig by ψ n − l ig ( a ⊗ b , . . . , a n − l ig ⊗ b n − l ig ) := τ n − l ig ( a , . . . , a n − l ig ) φ ( b · · · · · b n − l ig ) . This cocycle extends to a linear form on C n − l ig +1 ( 𝒜 h g i n − l ig ,l ig ) , which we again denote by ψ n − l ig . With the help ofit we can locally define on ind GH U α an analogous to (26) composition of maps which yields the wanted generalizedtrace density χ gi : b 𝒞 • ( 𝒟 X ⋊ G ) | ind GH U α → (cid:16) j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi (cid:17) G | ind GH U α ( D g , . . . , D p g p ) X k ≥ ( − ⌊ k ⌋ ψ n − l ig (( b s ◦ π i h g i , . . . , b s p ◦ π i h g i ) × ( ω α ) k ) where b s , . . . , b s p are the horizontal sections in j gi ∗ π coor ∗ 𝒪 flat ( 𝒩 coor × 𝒜 h g i n − l ig ,l ig ) corresponding to D g , . . . , D p g p . Alengthy calculation similar to (27) shows that by the GL n − l ig ( C ) × Z -basicness of ψ n − l ig the so-defined map isindependent on the choice of ind GH U α and hence extends to a well-defined morphism of cochain complexes on the G -equivariant topology of X or equivalently X/G . χ gi : b 𝒞 • ( 𝒟 X ⋊ G ) → (cid:16) j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi (cid:17) G . This map can be seen as a trace density morphism of the Hochschild chain complex of 𝒟 X ⋊ G . We can definesimilar maps for all strata X iH in X but we are only interested in trace density morphism to strata associated tocyclic subgroups of G . Theorem 3.11.
For every choice of a linearly independent trace φ i h g i of b 𝒟 l ig ⋊ h g i and for every family of holomorphicMaurer-Cartan forms { ω α } on the cotangent bundle T ∗ X gi of X gi with values in 𝒜 h g i n − l ig ,l ig the map ⊕ i,g ∈ G χ gi : b 𝒞 • ( 𝒟 X ⋊ G ) → (cid:16) ⊕ i,g ∈ G j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi (cid:17) G (32) is a quasi-isomorphism. Moreover, the induced morphism at the level of homology sheaves is canonical.Proof. The zeroth Hochschild cohomology group of 𝒟 l ig ⋊ h g i with values in the dual bimodule ( 𝒟 l ig ⋊ h g i ) ∗ definesthe space of traces on 𝒟 l ig ⋊ h g i . From [AFLS00, Proposition 3.1] we infer the isomorphism r : HH ( 𝒟 l ig ⋊ h g i , 𝒟 l ig ⋊ h g i ∗ ) ∼ = (cid:16) ⊕ ord( g ) k =1 HH ( 𝒟 l ig , 𝒟 l ig g k ∗ ) h g i (cid:17) where ord( g ) is the order of the generator g in G . As each group HH ( 𝒟 l ig , 𝒟 l ig g k ∗ ) h g i is spanned by the g k -twistedtrace Tr g k , defined in [Fed00], we have for the image of every trace φ i h g i of 𝒟 l ig ⋊ h g i , associated to the normal bundleto X gi , under r r ( φ i h g i ) = ord( g ) X k =1 λ k Tr g k ( · ) where for each k , the constant λ k is a complex number, including possibly zero. On the other hand, invoking thefact that 𝒟 X ⋊ G and 𝒟 X are sheaves of Calabi-Yau algebras of dimension n we have the natural identification s : ℋℋ • ( 𝒟 X ⋊ G ) ∼ = (cid:16) ⊕ g ∈ G ℋℋ • ( 𝒟 X , 𝒟 X g ) (cid:17) G hich combined with the induced map of (32) on homology yields the map ⊕ g,i χ gi ∗ ◦ s − : (cid:16) ⊕ g ∈ G ℋℋ • ( 𝒟 X , 𝒟 X g ) (cid:17) G → (cid:16) ⊕ i,g ∈ G H −• (cid:0) j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi (cid:1)(cid:17) G (33)where χ gi ∗ ◦ s − ( D g, D , . . . , D p ) = P i ,...,i p P k ( − ⌊ k ⌋ ˜ ψ n − l ig ( b D ,i ⊗ b D ⊥ ,i g, . . . , b D p,i p ⊗ b D ⊥ p,i p ) with ˜ ψ n − l ig ( a ⊗ b , . . . , a n − l ig ⊗ b n − l ig ) = τ n − l ig ( a , . . . , a n − l ig ) r ( φ i h g i )( b · · · b n − l ig ) ,a i ⊗ b i ∈ 𝒜 h g i n − l ig ,l ig . If we denote by p the projection of X onto the orbifold quotient X/G , then every point ¯ x on X/G is the image ¯ x = p ( x ) of an element x on X with stabilizer some subgroup H of G . Consequently, we get forthe stalk of 𝒟 X ⋊ G at ¯ x ( 𝒟 X ⋊ G ) ¯ x = lim −→ p − ( ¯ V ) ∋ p − (¯ x ) 𝒟 X ( p − ( ¯ V )) ⋊ G = 𝒟 X | ind GH { x } ⋊ G where ind GH { x } := ` l ∈ G/H l { x } is the closed set of G -translates of the single point x in X . Morphism (32) is aquasi-isomorphism if and only if for every ¯ x in X/G , the stalk of (33) is an isomorphism. For the left hand side ofthe induced map on the homology we get ℋℋ • ( 𝒟 X ⋊ G ) ¯ x ∼ = HH • (( 𝒟 X ⋊ G ) ¯ x ) ∼ = (cid:16) ⊕ g ∈ G HH • ( 𝒟 X | ind GH { x } , 𝒟 X | ind GH { x } g ) (cid:17) G ∼ = (cid:16) ⊕ l ∈ G/H ⊕ g ∈ lHl − HH • ( 𝒟 X,lx , 𝒟 X,lx g ) (cid:17) G ∼ = (cid:16) C G ⊗ C H ⊕ g ∈ H HH • ( 𝒟 X,x , 𝒟 X,x g ) (cid:17) G ∼ = (cid:16) ⊕ g ∈ H HH • ( 𝒟 X,x , 𝒟 X,x g ) (cid:17) H = ⊕ C H ( g ) ∈ Conj( H ) (cid:16) HH • ( 𝒟 X,x , 𝒟 X,x g ) (cid:17) Z H ( g ) where Conj( H ) is the set of conjugacy classes in H , C H ( g ) is the conjugacy class of g in H and Z H ( g ) is thecentralizer group of g in H , respectively. Similarly, we get for the stalk of the homology sheaf at ¯ x on the right handside of (32) (cid:16) ⊕ i,g ∈ G H −• (cid:0) j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi (cid:1)(cid:17) G ¯ x ∼ = ⊕ C H ( g ) ∈ Conj( H ) H −• ( j gi ∗ π gi ∗ Ω n − l ig −• T ∗ X gi ,x ) Z H ( g ) ∼ = ⊕ C H ( g ) ∈ Conj( H ) H n − l ig −• ( π gi ∗ Ω −• T ∗ X gi ,x ) Z H ( g ) ∼ = ⊕ C H ( g ) ∈ Conj( H ) H n − l ig −• (Ω −• X gi ,x ) Z H ( g ) ∼ = ⊕ C H ( g ) ∈ Conj( H ) H n − l ig −• ( 𝒜 −• X gi , C ,x ) Z H ( g ) where we dropped the index i , because a point cannot simultaneously be on more than one connected component ofa fixed point submanifold, and in the third line we used the holomorphic Poincare’s Lemma and the last isomorphismfollows from the fact that 𝒜 • X, C and Ω • X are both resolutions of the constant sheaf C X . We know from [FT10] thatfor every g in G , the cohomology group HH • ( 𝒟 X,x , 𝒟 X,x g ) is spanned by the cocycle c n − l ig ,x := X σ ∈ S n − lig ⊗ u σ (1) ⊗ · · · ⊗ u σ (2 n − l ig ) with u j − = ∂ x j and u j = x j . Then, the normalisation property of the Feigin-Felder-Schoikhet cocycle implies χ i h g i ,x ( s − ( c n − l ig ,x )) = ( − ⌊ n − l ig ⌋ ψ n − l ig ( s − ( c n − l ig ,x ))= ( − ⌊ n − l ig ⌋ ord( g ) X k =1 λ k τ n − l ig ( c n − l ig ,x ) Tr g k (1) ( − ⌊ n − l ig ⌋ ord( g ) X k =1 λ k Tr( g k ) = 0 which means that for each g , χ i h g i ,x ∗ ◦ s − x is a non-zero map between the generators of the -dimensional homologygroups HH • ( 𝒟 X,x , 𝒟 X,x g ) Z H ( g ) and H n − l ig −• ( 𝒜 −• X h g i , C ,x ) Z H ( g ) . Hence, it is an isomorphism. Thus, χ i h g i ,x ∗ andcorrespondingly the direct sum ⊕ i,g ∈ G χ i h g i ,x ∗ are invertible maps. Ergo, Morphism (32) is a quasi-isomorphism.Moreover, by normalization the stalk of the map (32) can be made independant on the particular choice of a trace φ i h g i . Ergo, Morphism (32) induces a canonical isomorphism at the level of homology. (cid:3) Composing the quasi-isomorphisms from Corollary 3.8 and (32) one obtains a new quasi-isomorphism which canbe interpreted as a generalized trace density morphism for the Hochschild cochian complex of 𝒟 X ⋊ G . To avoidrepetition we leave the standard proof of the next result to the interested reader. Corollary 3.12.
There is a quasi-isomorphism 𝒳 : b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) → (cid:16) ⊕ i,g ∈ G j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi (cid:17) G . (34) The induced morphism at the level of cohomology sheaves is canonical.
The space of filtered infinitesimal deformations of 𝒟 X ⋊ G . In the case of the Calabi-Yau algebras ( 𝒟 X ) G and 𝒟 X ⋊ G the inclusion of the subcomplex of filtration preserving Hochschild cochains into the complex of allHochschild cochains is a quasi-isomorphism. We only show this for 𝒟 X ⋊ G as the proof for ( 𝒟 X ) G is analogous. Proposition 3.13.
The canonical inclusion i : b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) ֒ → b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) is a quasi-isomorphism.Proof. Denote by X × k the k -fold Cartesian product of X . Let δ k : X → X × k , x ( x, . . . , x ) be the diago-nal embedding of X in X × k . Recall from the definition of the exterior tensor product that δ ∗ k 𝒟 X × k ∼ = 𝒟 ˆ ⊗ kX and δ ∗ k Sym • ( 𝒯 X × k ) = Sym • ( 𝒯 X ) ˆ ⊗ k . By definition the nonnegative decreasing filtration of b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) results in a first quadrant spectral sequence E pqr with zeroth sheet E pq := Gr p b 𝒞 p + qf ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) ∼ = { F ∈ ℋℴ𝓂 C (Gr q ( 𝒟 X ⋊ G ˆ ⊗ n ) , Gr q − p ( 𝒟 X ⋊ G )) | for all q ≥ p and all n ≥ }∼ = { F ∈ ℋℴ𝓂 C (Gr q ( δ ∗ n 𝒟 X × · · · × X | {z } n times ) ⋊ G × · · · × G | {z } n times , Gr q − p ( 𝒟 X ) ⋊ G ) | for all q ≥ p and all n ≥ }∼ = { F ∈ ℋℴ𝓂 C ( δ ∗ n Sym q ( 𝒯 X × · · · × X | {z } n times ) ⋊ G × · · · × G | {z } n times , Sym q − p ( 𝒯 X ) ⋊ G ) | for all q ≥ p and all n ≥ }∼ = { F ∈ ℋℴ𝓂 C ( (cid:2) Sym • ( 𝒯 X ) ˆ ⊗ n (cid:3) q ⋊ G × · · · × G, Sym q − p ( 𝒯 X ) ⋊ G ) | for all q ≥ p and all n ≥ }∼ = { F ∈ ℋℴ𝓂 C ( (cid:2) Sym • ( 𝒯 X ) ⋊ G ˆ ⊗ n (cid:3) q , Sym q − p ( 𝒯 X ) ⋊ G ) | for all q ≥ p and all n ≥ }∼ = (cid:2) b 𝒞 p + q (Sym • ( 𝒯 X ) ⋊ G, Sym • ( 𝒯 X ) ⋊ G ) (cid:3) p with p + q = n where in the third and the fifth line we used the definition of a topologically completed tensorproduct, and (cid:2) · (cid:3) q denotes the homogeneous part of (homological) degree q of the respective graded algebra. Toshorten in the following the notation denote by ∆ G the image of the diagonal homomorphism ∆ : G → G × G .The first sheet of the spectral sequence is accordingly becomes E pq = H p + q (Gr p b 𝒞 • f ( 𝒟 X ⋊ , 𝒟 X ⋊ G )) ∼ = (cid:2) H p + q ( b 𝒞 • (Sym • ( 𝒯 X ) ⋊ G, Sym • ( 𝒯 X ) ⋊ G )) (cid:3) p = (cid:2) ℰ𝓍𝓉 • Sym • ( 𝒯 X ) e ⋊ G × G (Ind Sym • ( 𝒯 X ) e ⋊ G × G Sym • ( 𝒯 X ) e ⋊ ∆ G Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) ⋊ G ) (cid:3) p = (cid:2)(cid:0) ⊕ g ∈ G ℰ𝓍𝓉 • Sym • ( 𝒯 X ) e (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) · g ) (cid:1) G (cid:3) p = (cid:16) ⊕ g ∈ G (cid:2) ℋℋ p + q (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) g ) (cid:3) p (cid:17) G ∼ = (cid:16) ⊕ g ∈ G (cid:2) ℋℋ n − p − q (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) · g ) (cid:3) p (cid:17) G where in the fourth line we used [KS94, Proposition 2.2.9] and in the last line we used that Sym • ( 𝒯 X ) is a sheafof Calabi-Yau algebras with dimension n as per Proposition 3.5.We proceed as in [DE05, Proposition 8] to furthersimplify E pq . There is a natural holomorphic splitting of the holomorphic cotangent bundle T ∗ X = T ∗ X g ⊕ N g ,where N g is the normal bundle to the fixed point submanifold X g . With the help of the resolution (22) of Sym • ( 𝒯 X ) ,we get for the homology sheaf of b 𝒞 • (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) · g ) ℋℋ • (Sym • ( 𝒯 X ) , Sym • ( 𝒯 X ) · g ) = 𝒯ℴ𝓇
Sym • ( 𝒯 X ) e • (cid:0) Sym • ( 𝒯 X ) · g, Sym • ( 𝒯 X ) (cid:1) = H • (Sym • ( 𝒯 X ) · g ⊗ Sym • ( 𝒯 X ) e Sym • ( 𝒯 X ) ⊗ C Sym • ( 𝒯 X ) ⊗ δ − 𝒪 Y | ∆ • ^ δ − 𝒯 ∗ Y | ∆ ) ∼ = H • (Sym • ( 𝒯 X ) · g ⊗ δ − 𝒪 Y | ∆ • ^ δ − 𝒯 ∗ Y | ∆ )= Sym • ( 𝒯 X g ) ⊗ δ − 𝒪 Y g | ∆ •− l g ^ δ − 𝒯 ∗ Y g | ∆ (35)where the complex in last line of (35) is equipped with the vanishing differential and l g is the complex codimensionof X g in X . The step from the second to the last line to the last line in (35) is a sheaf theoretic version of [Ann05,Proposition 4]. Making use of the relation Ω Y ∼ = 𝒯 ∗ Y we get(36) E pq = (cid:16) ⊕ g ∈ G Sym • ( 𝒯 X g ) ⊗ δ − 𝒪 Y g | ∆ δ − n − l g − ( p + q ) ^ 𝒯 ∗ Y g | ∆ (cid:17) G . As E pqr is a first quadrant spectral sequence, it converges to ℋℋ • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) . On the other hand thespectral cohomological sequence E pqr collapses on the p -axis, as shown in the picture below pq • E • E • E Accounting for the fact that the differential d pq : E pq → E p +1 ,q is the one of cohomological degree − , obtainedfrom the Koszul resolution of Sym • ( 𝒯 X ) , we obtain(37) E p := (cid:16) ⊕ g ∈ G H − p (cid:0) Sym • ( 𝒯 X g ) ⊗ δ − 𝒪 Y g | ∆ δ − n − l g −• ^ 𝒯 ∗ Y g | ∆ (cid:1)(cid:17) G . Combining the convergence of E pqr with (37) yields(38) E ∞ p = (cid:16) ⊕ g ∈ G H − p (cid:0) Sym • ( 𝒯 X g ) ⊗ δ − 𝒪 Y g | ∆ δ − Ω n − l g −• Y g | ∆ (cid:1)(cid:17) G ∼ = ℋℋ pf ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) . On the other hand, the filtration of 𝒟 X ⋊ G induces a bounded below and exhaustive filtration on the Hochschildchain complex b 𝒞 • ( 𝒟 X ⋊ G ) . This defines a spectral homological sequence E rpq with E pq := ℋℋ p + q (Sym • ( 𝒯 X ) ⋊ G ) ∼ = (cid:16) ⊕ g ∈ G Sym • ( 𝒯 gX ) ⊗ δ − 𝒪 Y g | ∆ p + q − l g ^ δ − 𝒯 ∗ Y g | ∆ (cid:17) G . ccording to the classical convergence theorem [Wei94, Theorem 5.5.1] this spectral homological sequence con-verges to ℋℋ p + q ( 𝒟 X ⋊ G ) ∼ = ℋℋ n − p − q ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) where the last isomorphism derives from the factthat 𝒟 X ⋊ G is a sheaf of Calabi-Yau algebras of dimension n . Obviously, E rpq collapses on the p -axis. Hence,(39) E ∞ p = (cid:16) ⊕ g ∈ G H p (Sym • ( 𝒯 gX ) ⊗ δ − 𝒪 Y g | ∆ •− l g ^ δ − 𝒯 ∗ Y g | ∆ ) (cid:17) G ∼ = ℋℋ n − p ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) From Isomorphism (38) and Isomorphism (39) we infer that ℋℋ pf ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) ∼ = ℋℋ p ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) for every p ≤ n . Since the map i is the canonical inclusion, at the level of cohomologies i ∗ remains injective. Weleave it to the reader to convince himself that the cohomology group sheaves of the complex of sheaves (cid:16) ⊕ g ∈ G Sym • ( 𝒯 gX ) ⊗ δ − 𝒪 Y g | ∆ • ^ δ − 𝒯 ∗ Y g | ∆ (cid:17) G are actually finite dimensional which implies that i ∗ is an isomorphism, as desired. (cid:3) Proposition 3.14. ˇH ( 𝒰 , σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G )) ∼ = (cid:16) ˇH ( 𝒰 , Ω ≥ X ) ⊕ L codim( X gi )=1 ˇH ( 𝒰 , j gi ∗ π gi ∗ Ω • T ∗ X gi ) (cid:17) G .Proof. We abuse notation by denoting an open cover of the orbifold
X/G and its preimage in the G -equivarianttopology of X by 𝒰 . Let for the sake of generality ℒ • denote an arbitrary complex of sheaves. Let ˇC • ( 𝒰 , ℒ • ) bethe Čech double complex thereof. Its total complex T • := Tot • (cid:0) ˇ C • ( 𝒰 , ℒ • (cid:1) has a natural decreasing filtration bythe second degree(40) F p T • := ⊕ n ⊕ i + j = nj ≥ p ˇ C i ( 𝒰 , ℒ j ) which determines a short exact sequence(41) → F p T • → F p − T • → F p − T • F p T • → for every p . Note the simple but important relation F p T • = ⊕ n ⊕ nj ≥ p ˇC n − j ( 𝒰 , ℒ j )= ⊕ n ⊕ j ≥ ˇC n − j (cid:0) 𝒰 , ( σ ≥ p ℒ ) j (cid:1) = Tot • (cid:0) ˇC • ( 𝒰 , σ ≥ p ℒ • (cid:1) . (42)The above defined filtration (40) applied to the total complex of the Čech double complex of (cid:16) ⊕ i,g ∈ G j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi (cid:17) G yields in account of (42) the short exact sequence of complexes → (cid:16) Tot • ( ˇC • ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X )) (cid:17) G ⊕ (cid:16) ⊕ i,g Tot • (cid:0) ˇC • ( 𝒰 , j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi ) (cid:1)(cid:17) G ℐ −→ (cid:16) ⊕ i,g Tot • (cid:0) ˇC • ( 𝒰 , j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi ) (cid:1)(cid:17) G 𝒫 −→ (cid:16) ˇC • ( 𝒰 , π ∗ 𝒪 T ∗ X ) (cid:17) G → . In turn it induces a long exact sequence of Čech hypercohomology groups · · · → (cid:16) ˇH ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X ) (cid:17) G ℐ ∗ −−→ (cid:16) ˇH ( 𝒰 , π ∗ Ω • T ∗ X ) (cid:17) G 𝒫 ∗ −→ (cid:16) ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) (cid:17) G∂ −→ (cid:16) ˇH ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X ) ⊕ ⊕ codim( X gi )=1 ˇH ( 𝒰 , j gi ∗ π gi ∗ Ω • T ∗ X gi ) (cid:17) G ℐ ∗ −−→ (cid:16) ˇH ( 𝒰 , π ∗ Ω • T ∗ X ) ⊕ ⊕ codim( X gi )=1 ˇH ( 𝒰 , j gi ∗ π gi ∗ Ω • T ∗ X gi ) (cid:17) G 𝒫 ∗ −→ (cid:16) ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) (cid:17) G → · · · (43)where ∂ denotes the so-called connecting morphism. A decomposition in terms of short exact sequences yields indegree the following short exact sequence → (cid:16) ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) (cid:17) G Im( 𝒫 ∗ ) ∂ −→ (cid:16) ˇH ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X ) ⊕ ⊕ codim( X gi )=1 ˇH ( 𝒰 , j gi ∗ π gi ∗ Ω • T ∗ X gi ) (cid:17) G ℐ ∗ −−→ ker( 𝒫 ∗ ) → . (44) n a similar fashion the short exact sequence → C X i −→ π ∗ 𝒪 T ∗ X p −→ π ∗ 𝒪 T ∗ X / C X → induces the long exactsequence · · · → ˇH ( 𝒰 , C X ) i ∗ −→ ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) p ∗ −→ ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) ∂ −→ ˇH ( 𝒰 , C X ) i ∗ −→ ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) p ∗ −→ ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) → · · · in which by abuse of notation ∂ again denotes the connecting morphism. It induces the short exact sequence → ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X )Im( i ∗ ) p ∗ −→ ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) ∂ −→ ker i ∗ → . (45)The pullback of the zero section s : X → T ∗ X defines a cochain map of sheaves s ∗ : π ∗ Ω • T ∗ X → Ω • X . By definition, s ∗ ◦ π ∗ = id . On the other hand a holomorphic homotopy operator K : π ∗ Ω pT ∗ X → Ω p − X can be constructedfollowing verbatim Chapter in [BT82] by means of which it can be shown that π ∗ ◦ s ∗ is cochain homotopic to theidentity of π ∗ Ω • T ∗ X . Thus, s ∗ is a quasi-isomorphism. Ergo, ˇH • ( 𝒰 , π ∗ Ω • T ∗ X ) ∼ = ˇH • ( 𝒰 , Ω • X ) . Consequently, fromthe fact that Ω • X is a resolution of C X we infer that Im( 𝒫 ∗ ) ∼ = Im( i ∗ ) and ker( 𝒫 ∗ ) ∼ = ker( i ∗ ) , respectively. Theisomorphism π ∗ 𝒪 T ∗ X / C X ∼ = π ∗ Ω T ∗ X, cl yields a morphism of Čech cochain complexes κ : ˇC • ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) → Tot • ˇC • ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X )[1] given by f mod C ( − n d dR ( f ) for every f mod C ∈ ˇC n ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) . Indeed, let D be the differential in Tot • ˇC • ( 𝒰 , π ∗ Ω • T ∗ X ) . Then, ¯ D n := ( − D n [1] := X p + q = n +1 − δ p,q − ( − p d p,qdR is the differential in degree n of the shifted total complex Tot • ˇC • ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X )[1] . Then, for every f mod C ∈ ˇC n ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) we have κ ( δ n ( f mod C )) = ( − n +1 d dR ( δ n ( f ))= |{z} δd dR − d dR δ =0 − δ n (( − n d dR ( f ))= − ( δ n + ( − n d dR )( − n d dR ( f )= ¯ D n (cid:0) κ ( f mod C ) (cid:1) . Subsequently, we fuse the short exact sequence (44) for the special case G = { id G } with the short exact sequence(45) in the single diagram ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X / C X )0 ˇH ( 𝒰 ,π ∗ 𝒪 T ∗ X )Im( i ∗ ) ker( i ∗ ) 0ˇH ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X ) κ ∗ ∂p ∗ ∂ ℐ ∗ which is commutative. Indeed, given an element f ∈ ˇC ( 𝒰 , π ∗ 𝒪 T ∗ X ) , for every α ∈ ˇC ( 𝒰 , π ∗ Ω T ∗ X ) , α = f + α ∈ Tot ˇC( 𝒰 , π ∗ Ω • T ∗ X ) satisfies 𝒫 ( α ) = f . Then, we have κ ∗ ◦ p ∗ ([ f ]) = κ ∗ ([ f mod C ])= [ d dR ( f )]= [ d dR ( f ) + D ( α )]= [ ℐ − ◦ D ( α )]= ∂ [ f ] . Similarly, by virtue of the above and the isomorphism ˇH ( 𝒰 , π ∗ Ω • T ∗ X ) ∼ = ˇH ( 𝒰 , C X ) , we have ℐ ∗ ◦ κ ∗ ([ f mod C ]) = [ D ( α )] ∂ [ f ] . By the -Lemma the linear morphism κ ∗ is in fact a linear isomorphism. This coupled to the fact that π ∗ 𝒪 T ∗ X / C X → π ∗ Ω ≥ T ∗ X [1] is a resolution of π ∗ 𝒪 T ∗ X / C X implies consequently(46) ˇH ( 𝒰 , σ ≥ π ∗ Ω • T ∗ X ) ∼ = ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X / C X ) ∼ = ˇH ( 𝒰 , π ∗ Ω ≥ T ∗ X ) ∼ = ˇH ( 𝒰 , Ω ≥ X ) . The morphisms 𝒳 and 𝒳 ≥ := σ ≥ ( 𝒳 ) induce correspondingly a quasi-isomorphism(47) ¯ 𝒳 : Tot • (cid:16) ˇC • (cid:0) 𝒰 , b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1)(cid:17) → (cid:16) ⊕ i,g Tot • (cid:0) ˇC • ( 𝒰 , j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi ) (cid:1)(cid:17) G and a morphism ¯ 𝒳 ≥ : Tot • (cid:16) ˇC • (cid:0) 𝒰 , σ ≥ b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1)(cid:17) → (cid:16) ⊕ i,g Tot • (cid:0) ˇC • ( 𝒰 , σ ≥ j gi ∗ π gi ∗ Ω •− l ig T ∗ X gi ) (cid:1)(cid:17) G , respectively.Let K •• be the Čech double complex of b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) associated to the cover 𝒰 : ... ... ... ˇC i +1 f ( 𝒰 , 𝒪 X ⋊ G ) · · · ˇC i +1 f (cid:0) 𝒰 , b 𝒞 jf ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1) ˇC i +1 f (cid:0) 𝒰 , b 𝒞 j +1 f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1) · · · ˇC if ( 𝒰 , 𝒪 X ⋊ G ) · · · ˇ C if (cid:0) 𝒰 , b 𝒞 jf ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1) ˇC if (cid:0) 𝒰 , b 𝒞 j +1 f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1) · · · ... ... ... δ d d δ d δ dδ d d δ d δ dδ δ δ in which δ denotes the Čech differential, d is the standard Hochschild differential with δd − dδ = 0 and in which bydefinition we have b 𝒞 f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) = ℋℴ𝓂 C ( C X , 𝒪 X ⋊ G ) ∼ = 𝒪 X ⋊ G . The total complex Tot • ( K •• ) hasa differential D ′ = δ + ( − p d in bidegree ( p, q ) .The natural filtration (40) of Tot • ( K •• ) yields in accordance with (41) and (42) the short exact sequence(48) → Tot • (cid:16) ˇC • (cid:0) 𝒰 , σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1)(cid:17) ℐ ′∗ −−→ Tot • ( K •• ) 𝒫 ′∗ −−→ ˇC • ( 𝒰 , 𝒪 X ⋊ G ) → . The canonical quasi-isomorphism i : b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) ֒ → b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) from Thereom 3.13 inducesa natural inclusion of Čech double complexes(49) ¯ i : ˇC • ( 𝒰 , b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G )) ֒ → ˇC • ( 𝒰 , b 𝒞 • ( 𝒟 X ⋊ G, 𝒟 X ⋊ G )) which yields an isomorphism at the level of Čech hypercohomologies. Combining (49) with the quasi-isomorphism(47), we obtain the maps p := ¯ 𝒳 ◦ ¯ ip ≥ := ¯ 𝒳 ≥ ◦ ¯ i With the help of p we arrive from (48) at the long exact sequence of Čech hypercohomology groups · · · → ˇH ( 𝒰 , σ ≥ b 𝒞 • f ( 𝒟 X , 𝒟 X )) ℐ ′∗ −−→ (cid:16) ˇH ( 𝒰 , π ∗ Ω • T ∗ X ) (cid:17) G 𝒫 ′∗ −−→ (cid:16) ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) (cid:17) G∂ ′ −→ ˇH ( 𝒰 , σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) ℐ ′∗ −−→ (cid:16) ˇH ( 𝒰 , π ∗ Ω • T ∗ X ) ⊕ ⊕ codim( X gi )=1 ˇH ( 𝒰 , j gi ∗ π gi ∗ Ω • T ∗ X gi ) (cid:17) G 𝒫 ′∗ −−→ (cid:16) ˇH ( 𝒰 , π ∗ 𝒪 T ∗ X ) (cid:17) G → · · · The brutally truncated map ¯ χ ≥ is not a quasi-isomorphism in general. omparison between the long exact sequence (43) and (48) delivers ker( 𝒫 ∗ ) = ker( 𝒫 ′∗ ) . The last fact allows us toextract from the above long exact sequence the following short exact sequence → (cid:16) ˇH ( 𝒰 , π ∗ 𝒪 X ) (cid:17) G Im( 𝒫 ∗ ) ∂ ′ −→ ˇH ( 𝒰 , σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G )) ℐ ′∗ −−→ ker( 𝒫 ∗ ) → . (50)By virtue of these morphism we can now merge the short exact sequences (44) and (50) in the ensuing diagram(51) ˇH2( 𝒰 ,σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G )) ˇH1( 𝒰 ,π ∗ 𝒪 T ∗ X ) G Im( 𝒫 ∗ ) ker( 𝒫 ∗ ) . (cid:16) ˇH2( 𝒰 ,σ ≥ π ∗ Ω • T ∗ X ) ⊕⊕ codim( Xgi )=1 ˇH0( 𝒰 ,jgi ∗ πgi ∗ Ω • T ∗ Xgi ) (cid:17) G p ≥ ∗ ℐ ′∗ ∂ ′ ∂ ◦ p ℐ ∗ In order to show that diagram (51) commutes, consider the diagram(52) Tot • (cid:16) ˇC • (cid:0) 𝒰 ,σ ≥ b 𝒞 • f ( 𝒟 X ⋊ G, 𝒟 X ⋊ G ) (cid:1)(cid:17) Tot • ( K •• ) ˇC • ( 𝒰 , 𝒪 X ⋊ G ) Tot • (cid:0) ˇC • (cid:0) 𝒰 , (cid:0) ⊕ i,g ∈ Gσ ≥ jgi ∗ πgi ∗ Ω •− ligT ∗ Xgi (cid:1) G (cid:1)(cid:1) Tot • (cid:0) ˇC • (cid:0) 𝒰 , (cid:0) ⊕ i,g ∈ Gjgi ∗ πgi ∗ Ω •− ligT ∗ Xgi (cid:1) G (cid:1)(cid:1) ˇC • ( 𝒰 ,π ∗ 𝒪 T ∗ X ) G . p ≥ ℐ ′ p 𝒫 ′ p ℐ 𝒫
The left-hand square of the diagram commutes because ℐ and ℐ ′ are the canonical inclusions. From this we im-mediately infer that the right-hand square of diagram (51) is commutative, too. On the other hand the right-handsquare in diagram (52) commutes, since for any β ∈ Tot • ( K •• ) , we have p ◦ 𝒫 ′ ( β ) = p ( β mod Im( ℐ ′ ))= p ( β ) mod Im( p ◦ ℐ ′ )= p ( β ) mod Im( ℐ ◦ p ≥ )= 𝒫 ◦ p ( β ) . (53)where in the third line we used the commutativity of the left square in Diagram (52). Finally, let the cohomologyclass [ f ] be an arbitrary representative in ˇH ( 𝒰 ,π ∗ 𝒪 T ∗ X ) G Im( 𝒫 ∗ ) . There is an element α ∈ Tot • ( K •• ) such that 𝒫 ′ ( α ) = f .Then, invoking the definition of the connecting morphism ∂ ′ , we obtain p ≥ ∗ ◦ ∂ ′ ([ f ]) = p ≥ ∗ ◦ [ ℐ ′− D ′ ( α )]= [ p ≥ ◦ D ′ ( α )]= [ D ( p ≥ ( α ))]= [ ℐ − ◦ D ( p ≥ ( α ))]= ∂ ◦ p ([ f ]) where in the second line we used that ℐ ′ is the canonical inclusion map and D ′ ( α ) ∈ Im( ℐ ′ ) , in the third the factthat p ≥ is a map of cocoain complexes and in the last line we used Equality (53). We conclude that the left-handside of diagram (51) commutes, whence the whole diagram is commutative. Then by the -Lemma, the map p ≥ ∗ isa linear isomorphism. This combined with Isomorphism (46) proves the claim. (cid:3) Theorem 2.10 coupled with Proposition 3.14 yields the following important isomorphism.
Corollary 3.15.
𝒟ℯ𝒻 ( 𝒟 X ⋊ G ) f ∼ = H ( X, Ω ≥ X ) G ⊕ (cid:0) L codim C ( X gi )=1 H ( X gi , C ) (cid:1) G as C -vector spaces. Since f is by definition a cocycle, D ( α ) ∈ ker( 𝒫 ′ ) = Im( ℐ ′ ) . roof. The only non-trivial part in the statement is the identification of ˇH ( X, j gi ∗ π gi ∗ Ω • T ∗ X gi ) with H ( X gi , C ) . Wedemonstrate that now. Let 𝒰 denote a G -invariant open cover of X and let the notation X gi ∩ 𝒰 on X gi denote theinduced open cover on X gi . With this, we have ˇH • ( X, j h g i∗ π gi ∗ Ω • T ∗ X gi ) = lim −→ 𝒰 ˇH • ( 𝒰 , j h g i∗ π gi ∗ Ω • T ∗ X gi )= lim −→ 𝒰 ˇH • ( X gi ∩ 𝒰 , π gi ∗ Ω • T ∗ X gi ) ∼ = lim −→ 𝒰 ˇH • ( X gi ∩ 𝒰 , 𝒜 • X gi , C ) ∼ = H • ( X gi , 𝒜 • X gi , C ) ∼ = H • ( X gi , C ) where in the third isomorphism we applied Poincare’s Lemma and the fact that 𝒜 • X gi and Ω • X gi are both resolutionsof C . (cid:3) We arrive at the main theorem of the paper.
Theorem 3.16.
Let X be a smooth algebraic variety or a smooth analytic variety equipped with a finite subgroup G ⊂ Aut( X ) acting faithfully on X . The sheaf of twisted Cherednik algebras ℋ ,c,ψ,X,G on the quotient orbifold X/G with formal c and ψ is a universal formal filtered deformation of the sheaf of filtered skew-group algebra 𝒟 X ⋊ G .Proof. The claim follows from the fact that the dimension of the parameter space { ( ψ, c ) } is the same as the dimen-sion of 𝒟ℯ𝒻 ( 𝒟 X ⋊ ) f . (cid:3) Acknowledgement
I would like to express my gratitude to the hospitality of the Department of Mathematics at MIT and the excellentresearch conditions which enabled my work at first place. I would like to extend my sincere gratitude to Prof. PavelEtingof for multiple valuable comments and suggestions which he generously provided in the course of my workon this note and which substantially improved its quality. In particular, I thank him for giving me the idea to usefiltration preserving Hochschild cochains- a trick which "unlocked" the whole chain of arguments in the note. I alsothank Prof. Giovanni Felder, Prof. Ajay Ramadoss, Prof. Valery Lunts and Dr. Konstantin Jacob for a useful exchangeand valuable comments on various topics. This work is financially supported by a
Swiss National Science FoundationEarly Postdoc.Mobility Fellowship number 188014.
References [AFLS00] J. Alev, M.A. Farinati, T. Lambre, and A.L. Solotar. “Homologie des invariants d’une algÚbre de Weylsous l’action d’un groupe fini”. In:
Journal of Algebra issn : 0021-8693.[Ann05] Rina Anno. “Multiplicative structure on the Hochschild cohomology of crossed product algebras”. In: arXiv preprint math/0511396 (2005).[BR73] I. N. Bernshtein and B. I. Rozenfe’ld. “Homogeneous spaces of infinite-dimensional Lie algebras and thecharacteristic classes of foliations”. In:
Uspehi Mat. Nauk issn : 0042-1316.[BT82] Raoul Bott and Loring W Tu.
Differential forms in algebraic topology . Springer, 1982.[DE05] Vasiliy Dolgushev and Pavel Etingof. “Hochschild cohomology of quantized symplectic orbifolds and theChen-Ruan cohomology”. In:
International Mathematics Research Notices doi : .[DMZ07] Martin Doubek, Martin Markl, and Petr Zima. “Deformation theory (lecture notes)”. In: Archivum math-ematicum
Ann. Sci. Éc. Norm. Supér.(4) arXiv preprintmath/0406499 (2004).[Fed00] Boris Fedosov. “On G-Trace and G-Index in Deformation Quantization”. In:
Letters in MathematicalPhysics issn : 1573-0530. doi : . FT10] Giovanni Felder and Xiang Tang. “Equivariant Lefschetz number of differential operators”. In:
Mathema-tische Zeitschrift arXiv preprint math/0612139 (2006).[Gra61] John W. Gray. “Extensions of sheaves of algebras”. In:
Illinois J. Math. doi : .[KS94] Masaki Kashiwara and Pierre Schapira. Sheaves on manifolds . Vol. 292. Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of Mathematical Sciences]. With a chapter in French byChristian Houzel, Corrected reprint of the 1990 original. Springer-Verlag, Berlin, 1994, pp. x+512. isbn :3-540-51861-4.[Mac75] Saunders MacLane.
Homology . Classics in Mathematics. Springer-Verlag, 1975.[Ram11] Ajay C Ramadoss. “Integration of cocycles and Lefschetz number formulae for differential operators”. In:
Symmetry, Integrability and Geometry: Methods and Applications Z -singularity”. In: International Mathematics Research Notices
Proceedings of the American Mathematical Society arXiv (Jan. 2019).[Vit20] Alexander Vitanov. “Trace densities and Index Theorems for the sheaf of Cherednik algebras”. In: arXiv (June 2020).[Wei94] Charles A. Weibel.
An Introduction to Homological Algebra . Cambridge Studies in Advanced Mathematics.Cambridge University Press, 1994. doi : .[Wod87] Mariusz Wodzicki. “Cyclic homology of differential operators”. In: Duke Math. J. doi : . Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave.,Cambridge, MA 02139, USAE-mail address : [email protected]@mit.edu