Chern classes of quantizable coisotropic bundles

Let $M$ be a smooth algebraic variety of dimension $2(p+q)$ with an algebraic symplectic form and a compatible deformation quantization $\mathcal{O}_h$ of the structure sheaf. Consider a smooth coisotropic subvariety $j: Y \to M$ of codimension $q$ and a vector bundle $E$ on $Y$. We show that if $j_* E$ admits a deformation quantization (as a module) then its characteristic class $\widehat{A}(M) exp(-c(\mathcal{O}_h)) ch(j_* E)$ lifts to a cohomology group associated to the null foliation of $Y$. Moreover, it can only be nonzero in degrees $2q, \ldots, 2(p+q)$. For Lagrangian $Y$ this reduces to a single degree $2q$. Similar results hold in the holomorphic category. This is a companion paper of a joint work with Victor Ginzburg on general quantizable sheaves.


Introduction
Let (M, ω) be a smooth variety of dimension 2(p + q) over a field k of characteristic zero, with an algebraic symplectic form ω (or corresponding holomorphic objects over k = C).We assume that the structure sheaf O M admits a compatible deformation quantization O h and fix a choice of such quantization.In other words, O h is a sheaf (in the Zariski or analytic topology, respectively) of complete, separated and flat k [[h]] algebras such that O h /hO h ≃ O M and if a → a 0 is the quotient map then a * b − b * a = hP (da 0 , db 0 )(mod h 2 ) where P ∈ H 0 (M, Λ 2 T M ) is the Poisson bivector corresponding to the symplectic form ω under the isomorphism Consider a coherent sheaf E h of O h -modules which is complete, separated and flat over k [[h]].
See [KS] for a general overview of modules over deformation quantization.We view E h as a quantization of its "principal symbol" σ(E h ) = E h /hE h , a coherent sheaf of O M -modules.A broad, but difficult, question is to establish necessary and sufficient conditions which would imply existence of E h .One way to simplify the situation is to assume that the support Y j → M of E h is smooth, and in fact E h /hE h is a direct image j * E of a locally free sheaf E on Y .
A straigtforward observation, which we recall below, is that in this case Y should be coisotropic, i.e. if N is the normal bundle of Y in M then the projection of P to H 0 (Y, Λ 2 N ) should be zero.Then p = 1 2 (dim M − 2 codim Y ) is a non-negative integer.When p = 0, i.e.Y is Lagrangian, papers [BGKP] and [BC] establish necessary and sufficient conditions for existence of E h .First, the associated projective bundle P(E) on Y should admit a flat algebraic connection.In particular, the Chern character of E equals e • exp(c 1 (E)) with e = rk E. To formulate the remaining conditions, recall that a choice of O h induces the Deligne-Fedosov class c(O h ) ∈ 1 h H 2 DR (X) [[h]] of the form Note that our present indexing of the coefficients is shifted by 1 as compared to that of [BC].
The Lagrangian property of Y implies that [ω] restricts to zero on Y .In [BGKP], [BC] it was shown that existence of quantization also implies ω i | Y = 0, for the cohomology classes ω i with i ≥ 1.This may be viewed as a strengthened Lagrangian condition which depends on the choice of O h .As for the cohomology class ω 0 , it admits a canonical lift (depending on O h ) to H 2 DR (Y, Ω ≥1 ) which is involved in the equation: where K Y is the canonical bundle of Y .Square root of the canonical class has appeared in the contact setting in [Ka].In the holomorphic setting, quantization of the square root of the canonical class is due to D'Agnolo and Schapira, see [DS].See also [Bo] and [NT2].
Returning to the case of a non-necessarily Lagrangian smooth coisotropic Y , define the "quantum Chern character" class considered in [NT1] (see also [BNT] and [CFW] for later results): where the A-genus is recalled in Section 2.2.We understand τ (E h ) as a class with values in the de Rham cohomology H * DR,Y (M )((h)) with support at Y , which can be identified with the de Rham cohomology of Y , due to smoothness.The purpose of this note is the following result Theorem 1.1 If the principal symbol sheaf σ(E h ) is isomorphic to the direct image j * E of a locally free sheaf E on a smooth coisotropic subvariety j : Y → M of codimension q then the class τ (E h ) ∈ H * DR,Y (M )((h)) is zero except in degrees 2q, . . ., 2(p + q) = dim k M .Moreover, if Ω F ⊂ Ω 1 Y is the sheaf of 1-forms that vanish on the null-foliation (or characteristic foliation) F ⊂ T Y and F r Ω • Y is the ideal in the de Rham complex generated by the r-th power of Ω F , then τ (E h ) is in the image of the map Our strategy is an application of formal geometry and the Gelfand-Fuks map: first use Riemann-Roch theorem to replace τ (E h ) by an element τ Y (E) ∈ H • DR (Y )((h)); then show that after completion both the quantized functions and the quantized module are isomorphic to standard objects and construct a Harish-Chandra torsor (foliated over F ) and a Lie algebra cohomology class that induces τ (E h ) via Gelfand-Fuks map.At this point the vanishing reduces to a vanishing in Lie algebra cohomology for which we use a Lie Algebraic version of the Index Theorem, cf.[NT1], [BNT], [CFW], [CD], [PPT], [GKN], [GLX], and that fact that in the symplectic situation the trace map can be defined on negative cyclic homology.
Alternatively, one could use an observation due to B. Tsygan that the Chern character of a perfect complex factors through negative cyclic homology of its derived endomorphism algebra, but in the algebraic geometry setting the Lie algebra cohomology route seems a bit shorter.
Remarks.(i) In a forthcoming paper with V. Ginzburg, cf.[BG], we prove a similar statement for quantizable sheaves with arbitrary support, including the fact that τ (E h ) agrees with the general Connes' Chern character and that the Algebraic Index Theorem holds for general algebraic varieties.Since in general a formal completion of a quantized sheaf will not be isomorphic to a "standard formal model", methods of formal geometry do not apply for general sheaves.
(ii) It would be very interesting to relate our main theorem to Bordemann's criterion for existence of second order quantization (modulo h 3 ).This might depend on what can be said about the Atiyah-Molino class of the charactersitic foliation of Y .See [Bo] for more details.
The paper is organized as follows.In Section 2 we recall standard constructions related to foliations, characteristic classes and use Riemann-Roch theorem to reduce the main result to a cohomology class on Y .In Section 3 we recall definitions related to Harish Chandra pairs and torsors and the Gelfand Fuks map.We further state the Lie cohomology Algebraic Index Theorem and prove a vanishing result for the class involved.The conceptual reason for the vanishing is that Connes Chern character with values in periodic cyclic homology lifts to negative cyclic homology.In Section 4 we prove the main result by constructing two Harish-Chandra torsors that induce the class under consideration, and then invoking the vanishing of Section 3.
Acknowledgements.The author thanks V. Ginzburg, J. Pecharich and T. Chen for the useful conversations.This work was supported by the Simons Collaboration Grant #281515.
2.1 Null foliation and a filtration on the de Rham complex.
We start by assuming that a pair (O h , E h ) is given as in the introduction and that σ(E h ) = E h /hE h is the direct image j * E of a locally free sheaf supported on a smooth subvariety Y .We use the same notation E for associated vector bundle on Y .
If I ⊂ O M is the ideal sheaf of functions vanishing on Y and x ∈ O h is a local section projecting to I, then x • E h ⊂ hE h .If y is another such section, it follows that the commutator in O M also annihilates j * E, i.e belongs to I. Thus, the ideal sheaf I is closed with respect to the Poisson bracket induced by the Poisson bivector P .
If N is the normal bundle of Y in M , we can restate this by saying that P | Y projects to the zero section in H 0 (Y, Λ 2 N ), and then the same restriction defines a section in H 0 (Y, T Y ⊗ N ).We can view the latter as a morphism N ∨ → T Y and it is easy to check that it is an embedding of vector bundles.Using j * N ∨ ≃ I/I 2 we can write an explicit local formula for it: Denote by F ⊂ T Y the image of this embedding, i.e. the null-foliation of Y .By the above, this sub-bundle is involutive, i.e. closed with respect to the bracket of vector fields on Y (since the Poisson bracket on I/I 2 is compatible with the bracket on vector fields).
The involutive property can be restated as follows.Let Ω F = (T Y /F ) ∨ ⊂ Ω 1 Y be the sheaf of 1-forms vanishing along F and denote by Y the graded ideal generated by Ω F in the sheaf of differential forms on Y , viewed as a sheaf of graded commutative algebras.By a straightforward application of the formula the involutive property of F is equivalent to the statement that F 1 Ω • is a subcomplex of the de Rham complex.It follows that each power of the ideal The main message in this paper is that characteristic classes of interest lift to the cohomology groups H 2r (Y, F r Ω • ).Note that the rank of Ω F is 2p = dim Y − rk(F ) = dim M − 2 codim Y hence for r > 2p the relevant cohomology group vanishes as F r Ω • is the zero subcomplex.Moreover, the particular class τ (E h ) vanishes for r > p.
2.2 Riemann-Roch Theorem and reduction to a class on Y .
Recall that for a power series G(z) with constant term 1 and a vector bundle V of rank v we can define its multiplicative G-genus as a product G(z i ) where z 1 , . . .z v are the Chern roots of V , i.e. formal variables such that the l-th elementary symmetric function in z i is equal to the l-th Chern class c l (V ).Recall also that the A-genus A(V ) and the Todd genus T d(V ) correspond to respectively.The Todd genus is involved in the Grothendieck-Riemann-Roch theorem for a closed embedding j : Y → M and a coherent sheaf E on Y (see Section 15.2 of [Fu2]): where N is the normal bundle to Y .We use this formula to study the "quantum" class which appears in the Index Theorem of [BNT], [NT1] and the Local Index Formula of [CD].By Riemann-Roch and the Projection Formula we can rewrite this expression as Note that G 1 (z) is an even function of z hence A(N ) = A(N ∨ ).Using the multiplicative property of genus and the short exact sequences (where the second short exact sequence is the definition of Q) we conclude that Hence Theorem 1.1 reduces to the statement that the class ) and vanishes in for r > p.Note that when M = Y and M is projective the cohomology groups will be nonzero up to degree 4p = 2 dim M .
The purpose of this section is to review some known results involving Lie Algebra Cohomology and characteristic classes, as they apply to deformation quantization; and also to fix the notation.
3.1 Lie algebra cohomology and a version of the Chern-Weyl map.
For a Lie subalgebra h ⊂ g the subcomplex of relative Lie cochains, cf.Chapter 1.3 in [Fu1], C • (g, h; V ) give by the condition that both α and d Lie (α) vanish when one of their arguments is in h.Its cohomology groups are denoted by H • (g, h; V ).One source of relative Lie cocycles, see Section 2.2 of [NT1] arises from an ad(h)-invariant projection pr : g → h and its curvature Assume for simiplicity that the g action on V is trivial.Then for any h-invariant polynomial is relative with respect to h, closed, and its relative cohomology class is independent on the choice of the projection pr : g → h.This defines the Chern-Weil homomorphism We will need the following examples of relative cocycles: for y ∈ h.

For a central extension of Lie algebras
0 → a → g → g → 0 and a k-vector space splitting g ≃ a ⊕ g, the 2-cocycle C : Λ 2 g → a is the curvature as above.In the cases we consider the cocycle may be chosen in C 2 (g, h; a).

Torsors over Harish-Chandra pairs and characteristic classes.
Definition.A Harish-Chandra pair (g, F ) consists of a Lie algebra g, a (pro)algebraic group F over k, an embedding of Lie algebras f = Lie(F ) ⊂ g and an action of F on g which extends the adjoint action of F on f.A module over a Harish-Chandra pair (g, F ) is an F -module V with an F -equivariant Lie morphism g → End k (V ) extending the tangent Lie morphism on f.
In this paper, f will have finite codimension in g and F ≃ L ⋉ U with L a finite dimensional reductive group and U a pro-unipotent infinite dimensional algebraic group.
Definition.A Harish-Chandra torsor or a flat (g, F )-torsor over a scheme Y is an F -torsor π : P → Y with an F -equivariant g-valued 1-form γ : T P → g ⊗ k O P which restricts to the canonical Maurer-Cartan form (with values in f ⊂ g) on the vector fields tangent to the fibers of π : P → Y , and satisfies the Maurer-Cartan equation where d is the de Rham differential on P and the bracket is computed in g.In the infinite dimensional case some care must be taken to define such torsors.One possible approach is to follow the pattern in [Ye], [VdB] and work with representable functors.In our case, eventually we will only need direct images of differential forms from P to Y , and since F is a limit of affine groups, all geometric objects on Y can be defined using shaves on Y with a coaction of functions on F , and so on.See Sections 2 and 3 in [DGW] for a closely related case.
Assume that γ : T P → g ⊗ k O P is onto and has kernel T γ of finite constant rank q.Note that since γ is injective on vertical vector fields, the differential of π : P → X is injective on T γ .Moreover, since γ is F -equivariant, for points x, y in the same orbit of F , the images (T γ ) x and (T γ ) y in (T Y ) π(x)=π(y) agree.Let F ⊂ T Y the resulting rank q sub-bundle on Y .
Lemma 3.1 In the situation described, let T π = Ker(dπ) ⊂ T P be the vector fields tangent to the fibers.Then the sub-bundles F ⊂ T Y and G = T π ⊕ ker(γ) ⊂ T P are integrable (i.e.stable under the Lie bracket of vector fields).Let Ω F ⊂ Ω 1 Y be the annihilator of F and Ω G ⊂ Ω 1 P the annihilator of G. Denote by F r Ω • Y , resp.F r Ω • P , the graded ideal generated by the r-th power of Ω F , resp. the r-th power of Ω G .Then both F r Ω • Y and F r Ω • P are preserved by the corresponding de Rham differentials and there is a morphism of complexes of sheaves We start with G.If v 1 , v 2 are two vector fields in Ker(γ) then formula (2) for dγ and the Maurer-Cartan equation for γ imply that the bracket [v 1 , v 2 ] is also annihilated by γ.The fact that vector fields tangent to the fibers are closed with respect to the Lie bracket holds for any smooth π.Finally, let's assume that v 1 ∈ Ker(γ) and v 2 ∈ T π .Then the quadratic term in the Maurer-Cartan equation vanishes on v 1 ∧ v 2 (as γ(v 1 ) = 0) and we are left with The second term is zero by assumption on v 1 and the first is in In particular, a bracket of two F -equivariant vector fields in Ker(γ) is again an F -equivariant vector field in Ker(γ).Its F -equivariant descent is a rank q subbundle in the Atiyah algebra At P of P on Y (= the F -equivariant descent of all vector fields on P ), which is also closed under Lie bracket.It projects isomorphically to a sub-bundle F ⊂ T Y (as Ker(γ) has trivial intersection with T π ) which is closed with respect to the Lie bracket since At P → T P is compatible with brackets.By construction, 1-forms on Y which vanish on F pull back to F -equivariant 1forms on P which vanish on G. Hence the morphism of sheaves of dg algebras Ω • Y → π * Ω • P is compatible with multiplicative filtrations F r Ω • induced by the two foliations.
Definition.In the situation of the previous lemma we will say that the Harish-Chandar torsor (P, γ) is foliated over F ⊂ T Y .We will see that in this case some of its characteristic classes what apriori belong to Let V be a Harish Chandra module over (g, F ) with trivial action (as will be in our applications).
For a Lie l-cochain α : Λ l g → V the composition Λ l T P → Λ l g ⊗ k O P → V ⊗ k O P may be viewed as a V -valued l-form on P and the Maurer-Cartan equation ensures that the resulting Gelfand-Fuks morphism cf.Chapter 3.1.C,D in [Fu1] also agrees with differentials: We want to use this observation to study characteristic classes of P in the cohomology of Y .
Note that to obtain classes on Y we need to work with objects which are invariant with respect to the reductive subgroup of forms, i.e. forms β such that L v β = 0 and ι v β = 0 for any v ⊂ h (we use the same letter v for the vertical vector field on P induced by v via the action of F ).
Lemma 3.2 Assume that F = U ⋊ H is semi-direct product of a finite dimensional connected reductive group H and a pro-unipotent group U .If h = Lie(H), there exists a quasi-isomorphism of sheaves of dg-algebras If P is foliated over F ⊂ T Y then for any r ≥ 0 the natural morphism of ideal sheaves Sketch of Proof.First assume that U is finite dimensional and look at the first quasi-isomorphsm.Since H is connected, the pushforward of h-basic forms from P to Y may be identified with the pushforward of forms on P/H to Y .But P/H → Y is a bundle with affine fibers so the assertion follows from the relative Poincare Lemma (triviality of relative de Rham cohomology for fibrations by affine spaces).For infinite dimensional pro-unipotent U , we first consider the finite dimensional unipotent factors and then pass to a limit, as in Theorem 6.7.1 in [VdB] .
In view of the unfiltered quasi-isomorphism, its filtered version reduces to showing that the maps induced on associated graded quotients are quasi-isomorphisms.First, where Q = T Y /F and we consider Λ • F ∨ as a complex of sheaves with the differential similar to that of in formula (3) (in other words, it is the de Rham differential of the Lie algebroid F ⊂ T Y ).On the other hand, since the pullback of F is isomorphic to Ker(γ) ⊂ G and the pullback of Q is isomorphic to T P /G, by the projection formula we get where Ω • π is the relative de Rham complex of π : P → Y .Hence we just need to show that is a quasi-isomorphism, i.e. which again follows from the relative Poincare Lemma.
In view of the previous result, passing to cohomology in ( 5) we obtain an h-relative version of Gelfand-Fuks map, which we denote also by GF : Recall that whenever we use GF we assume that g-action on V is trivial, otherwise the right hand side would involve the de Rham cohomology of associate vector bundle V P with a flat connection induced by the Harish-Chandra module structure.
Proposition 3.3 For V = k in the setting of Lemma 3.2 the composition of the Lie algebraic Chern-Weil map and the Gelfand-Fuks map is the classical Chern-Weil map of the torsor P H , associated to P via the group homomorphism F → H.If P is foliated over F ⊂ T Y , for every r ≥ 0 the composition admits a canonical lift Proof.Let P U = P/H then P → P U maybe be viewed as the H-torsor pulled back from Y via P U → Y .Since P U → Y induces isomorphism on de Rham cohomology (relative Poincare Lemma) we can replace Y by P U and assume that U is trivial.Then the composition is a connection on P → P U and the Lie theoretic curvature The assertion follows since the classical Chern-Weil map may be computed by evaluating invariant polynomials on R ∇ .
For the second assertion, we observe that relative Lie cochains define a global section of π * (F r Ω • P ) h−basic and the result follows by application of Lemma 3.2.

Algebraic version of the Index Theorem.
Consider the formal Weyl algebra D p , the completion (at the augmentation ideal) of the universal enveloping of the Heisenberg Lie algebra with generators x 1 , . . ., x p , y 1 , . . ., y p , h and the only nontrivial commutators given by [y j , x i ] = δ ij h.In other words, as a vector space D p is isomorphic to k[[x 1 , . . ., x p y 1 , . . ., y p , h]] but with nontrivial commutation relations y i x i = x i y i + h.
We consider the associative algebra E = gl e (D p ) and the Lie algebra Der(E) of its continuous k[[h]]-linear derivations.Of course, some of the derivations are are inner and hence there is a Lie morphism E → Der(E).In addition, any commutator in D p is divisible by h, so commuting with a scalar 1 h D p -valued matrix also gives a derivation of D p .We claim a short exact sequence For e = 1 this is well-known, see e.g.Section 2.3 in [BNT] and equation (3.2) in [BK].For general e follows from the fact that the quotient of all derivations by inner derivations (i.e.first Hochschild cohomology group) is a Morita invariant, cf.Chapter 1.2 in [Lo].
The Lie algebra g = 1 h D p +E has a reductive subalgebra gl e ⊕sp 2n (matrices with values in k ⊂ D p plus an isomorphic copy of sp 2n in 1 h D p spanned by commutators of 1, 1 h x i x j , 1 h y i x j , 1 h y i y j ).We also consider the abelian subalgebra a To take into account a ∩ gl e = k, introduce a ′ ⊂ a with topological basis given by h is a Lie subalgebra of g.By Hochschild-Serre spectral sequence where V is a module over Der(E) (and thus also a module over g).Below, we are interested in the cohomology of Der(E) but find it easier to do computations in g.Now we would like to state a vanishing lemma for homogeneous components of a particular class in H • (g, h; k((h))).Its proof will take the rest of the section.A reader willing to treat it as a black box may wish skip to Section 4. We follow the notation and exposition in [GLX] which deals with a version of Algebraic Index Theorem that is most convenient for our setting.
This class can be defined via the Chern-Weil construction.To fix a projection pr : g → h we introduce a filtration on g by giving h degree 2 and x i , y j degree 1.Since elements of g involve infinite sums, g splits into a direct product i≥−2 g i .For the first two factors in h, we project g onto g 0 ≃ gl e ⊕ sp 2n (recall that sp 2n is spanned by the commutators in the degree zero part 1 h D p 0 ) and set k = h h k to be in the kernel on the projection onto sp 2n .For a ′ we choose any projection 1 h D p → a ′ and extend it by 0 to trace zero matrices in E.
Lemma 3.4 For h = gl e ⊕ sp 2p ⊕ a ′ , let ch Lie (gl e ), A Lie (sp 2p ) and C(a ′ ) be the classes induced by the Chern-Weil constrution at the end of Section 3.1, from the respective factors.Let Then the components of τ Dp of degree > 2p are equal to zero.
Proof.Our proof is based on the fact that the class of the theorem arises from the study of periodic cyclic homology of the associative algebra . The vanishing will follow from the fact that this class lifts to the negative cyclic homology of A. We briefly recall the relevant definitions here, omitting details that do not contribute to the proof.
. This is the cohomological grading, rather than homological grading used in some sources, although the indices are written as subscripts to avoid confusion with the Hochschild cohomology complex.The standard formulas e.g. in 1.1.1 and 2.1.8 of [Lo] define the Hochschild and cyclic differentials, that satisfy b 2 = 0, B 2 = 0, Bb + bB = 0. Introducing a formal variable u of cohomological degree 2, consider two complexes with cohomology defining the negative cyclic homology HC − • (A) and the periodic cyclic homology HC per • (A), respectively.In both cases 1 ∈ A = C 0 (A) satisfies (b + uB)(1) = 0 and thus gives a cohomology class.

In the case
, by the work of Shoikhet, cf.[Sh] and Section 1.2 in [CFW], there is a quasi-isomorphism (C • (A), b) → (Ω −• ((h)), hL π ) where Ω −• stands for formal differential forms in x 1 , . . ., x p , y 1 , . . ., y p and π = (∂/∂x i ) ∧ (∂/∂y i ) is the standard Poisson bivector.Furthermore, Willwacher proved in [Wi], that this quasi-isomorphism sends B to the de Rham differential.This results in a quasi-isomorphism On the left hand side, The explicit construction on the above quasi-isomorphism is quite non-trivial and involves integration over configuration spaces of points.The Lie algebra Der(E) of derivations of E = gl e (D p ) acts on periodic and negative cyclic complexes.It follows from the explicit construction in [Sh] that the quasi-isomorphsm is compatible with the action of pgl e (k) ⊕ sp 2n ⊂ Der(E) but not the full algebra of derivations.However, it can be upgraded to a cocycle in the relative Lie algebra cohomology of Der(E).As before, we work with g (a central extension of Der(E)) rather than derivations of A. Then the upgraded quasi-isomorphism is an element of total degree 0 We note here that the action of g is not Ω −• ((h)) is not h linear (in fact the element h ∈ g acts by zero, so the action factors through the quotient by h), and all cochains are only k-linear maps.This is usually considered for the periodic cyclic complex but we emphasize that at this step no inversion of u is necessary.Next, one can construct two g-invariant homomorphisms which are homotopic in the full Lie algebra cohomology complex.The first homomorphism needs inversion of u.We first use the identity e h u ιπ (hL π + ud DR ) = (ud DR )e h u ιπ to land in the complex ( Ω −• ((h))((u)), ud DR ).Then we apply the re-grading operator g u which is an isomorphism of complexes sending an i-form α to u −i α.Thus, on the left hand side α has cohomological degree (−i), on the right hand side degree i and adjustment by u −i makes the re-grading a degree 0 operator.
With these preliminaries, we now prove the vanishing claimed in Lemma 3.4 in several steps Step 1.First consider the class with τ Lie (we recall that the grading on differential forms has been inverted at this step).The range of indices is k ≥ 0 and 0 ≤ k +l ≤ n.Then move on to the class τ 2 = e − h u ιπ τ 1 which lives in the same complex but with the differential ud DR instead of hL π + ud DR .The components τ k,l 2 u l of the class τ 2 can be nonzero in the same range 0 ≤ k, 0 ≤ k + l ≤ n.
Step 2. Now consider the regraded class (where the differential forms now have the usual grading) and observe that the exponents of u are in the range (−2n, . . ., 0).The next step is to replace the de Rham complex ( Ω • , d DR ) by a quasiisomorphic complex (k, 0).Note that the projection Ω • → k (which vanishes on forms of positive degrees and sends a power series in degree zero to its constant term) -is not g-equivariant.
However, it can be extended to a quasi-isomorphism of complexes f : using Lemma 3.5 (i) below.This leads to a class Note that 0 ≤ k, 0 ≤ k + l ≤ n imply 0 ≤ 4k + 2l ≤ 4n.This agrees with the fact that the de Rham cohomology of M is nonzero in the range (0, . . ., 4n).Now we use the Algebraic Index Theorem in the form proved in [GLX] to claim that τ 4 = τ Dp , the class introduced in the lemma.
Step 3. We are finally in position to prove that on the level of cohomology the coefficients τ 4k+2l 4 vanish when 4k + 2l > 2n.Indeed, instead of which has trivial coefficients of u −(2k+l) , 2k + l ≥ n since the only negative power of u created is the factor u −n in the definition of g h .
Step 4. To show that τ 3 and τ ′ 3 have the same cohomology class we use the identity which can either be established by direct computation, or by using a basic observation of Hodge Theory that operators ∧ω and ι π generate an sl 2 action on the de Rham complex, hence the above identity can be obtained as the image of the group level identity in SL 2 under the action homomorphism.
Finally, we note that e ω uh is homotopic to identity.This follows from the fact that ω = d DR (α) where α = 1 2 (x i dy i − y i dx i ) is the Euler vector field converted to 1-forms using ω.Therefore we have d DR ϕ + ϕd DR = e ω uh where Again, the homotopy does not agree with the g-action (only with the h-action) hence we use Lemma 3.5(ii) below to obtain a homotopy between identity operator and e ω uh .
Hence we can use the above class τ ′ 3 instead of τ 3 .Since τ ′ 3 by construction has at worst poles of order ≤ n in the u variable, the assertion follows.
We finish here with a homotopy lemma used above Lemma 3.5 Let M • and N • two complexes of modules over a Lie algebra g and f : the first terms in the formula (3) and the combination of the second term of (3) with the internal differential on M • , N • , respectively.Let be the morphisms of complexes with d Hom differentials, induced by f, g, respectively, and ϕ Hom : Proof.Part (i) is a consequence of the Basic Perturbation Lemma, cf.[Ma] Part (ii) is easier to establish by direct computation although it is also a very degenerate case of the Ideal Perturbation Lemma, cf.[Ma].
4 Proof of the main result.
In this section we prove Theorem 1.1 by studying the characteristic class For y ∈ Y the preimage of the maximal ideal m y ⊂ O Y,y in the stalk at y, with respect to the reduction mod h map O h,y → O Y,y , is a maximal ideal m h,y ⊂ O h,y .Adapting the classical proof of the Darboux theorem, we show that after completion at this maximal ideal the triple ) is isomorphic -non-canonically!-to a similar triple independent of y or Y .Different choices of isomorphisms will give the Harish Chandra torsor P D,M inducing τ Y (E) via the Gelfand-Fuks map.Further, it is actually lifted from a quotient torsor P E and Theorem 1.1 will be reduced to the study of the class τ Dp where one uses Lemma 3.2 and Lemma 3.4.
4.1 Standard formal models: D, M and E.
Below for n = p + q we will assume that D q is the Weyl algebra built on the variables x i , y i , h, with i = 1, . . ., q, that D p corresponds to the values i = q + 1, . . ., q + p while D is the Weyl algebra on the full set of variables with i = 1, . . ., p + q = n.Fixing decomposition n = p + q and an integer e ≥ 1, define a left D-module (on the right hand side, we use completed tensor product).The second presentation implies the following isomorphism for endomorphisms of M (which we assume to be acting on the right ): Lemma 4.1 For any Y, O h , E h as before and y ∈ Y , denote by (. ..) the completion of a stalk at y with respect to the maximal ideal m h,y .
(1) There exist compatible isomorphisms as filtered algebras and modules, respectively.Moreover, if an algebra isomorphism σ D admits a compatible module isomorphism σ M then σ D sends the completion to the double sided ideal J ⊂ D generated by y 1 , . . ., y q , h.
(2) If σ D ( I h ) = J then σ D extends to a pair of compatible isomorphisms (σ D , σ M ).
Proof.Modulo h, we can construct an isomorphism of O M,y and k[[x 1 , . . ., x n , y 1 , . . .y n ]] since y is a smooth point, k has characteristic zero and X has dimension 2n.Since Y is smooth, its ideal I Y ⊂ O M,y is generated by a regular sequence and we can assume that the isomorphism sends the completion on I Y to the ideal generated by y 1 , . . ., y q .We can also adjust the isomorphism to be compatible with the symplectic forms.Let α be a two form on Spec(k[[x 1 , . . ., x n , y 1 , . . ., y n ]]) induced from ω via the initial isomorphism.Decompose it into homogeneous components: α = α 0 + α 1 + α 2 + . . .where each α i is a two form with coefficients of homogeneous degree i.Since Y is coisotropic, after a linear change of coordinates (x 1 , . . ., x n , y 1 , . . ., y n ) we can assume that α 0 = dx i ∧ dy i .
Note that the ideal J generated by (y 1 , . . ., y q ) is Poisson with respect to the bracket induced by α.This means that the coefficients of dx r ∧ dx s are in J for each α j and r, s ≤ q.We now want to find a formal vector field µ such that the formal diffeomorphism exp(µ) takes α to α 0 and preserves J .In fact we will construct exp(µ) inductively as the composition of exp(µ 1 ), exp(µ 2 ), . . .where µ i is a polynomial vector field with coefficients of homogeneous degree i and exp(µ i−1 ) . . .exp(µ 1 )(α) = α 0 + β ≥i with β ≥i a 2-form with coefficients of degree ≥ i.To ensure that J is preserved, we need to have µ i (J ) ⊂ J , which is to say, the coefficient of ∂/∂y r in µ i is an element of J for r ≤ q.Considering γ i = α 0 (µ i , •) we see that γ i needs to be a polynomial differential form with coefficients of degree i, such that the coefficient of dx r is in J for r ≤ q and dγ i = β i where β i is the degree = i component of β ≥i .
Note that β i is at least closed since this property holds for α, is preserved after the action of exp(µ j ) and β i is just a homogeneous component of the resulting form α 0 + β ≥i .Since the formal de Rham complex is exact in degrees ≥ 0, β i = dγ i with γ i = ι Eu β i , the contraction with the Euler vector field Eu = (x r ∂/∂x r + y r ∂/∂y r ).Note that by induction, after each formal diffeomorphism J remains a Poisson ideal hence the coefficient of dx r ∧ dx s in β i , is an element of J for r, s ≤ q.After the Euler field contraction, every coefficient of dx r in γ i is also in J , as required.Thus we have a formal diffeomorphism exp(µ i ) that will eliminate β i .
Passing to the formal limit i → ∞, we get an isomorphism of O M,y ≃ k[[x 1 , . . ., x n , y 1 , . . .y n ]] which takes the completion of I Y to the ideal J , and is compatible with the symplectic forms.This proves the "quasiclassical" part of the statement.
Both O h and D are deformation quantizations of the same algebra k[[x 1 , . . ., x n , y 1 , . . ., y n ]], corresponding to two formal Poisson bivectors h( ∂/∂x i ∧ ∂/∂y i ) + h 2 π 2 + . . .with the same h-linear part π 1 = ∂/∂x i ∧∂/∂y i .By the general Maurer-Cartan formalism, Poisson bivectors with fixed linear part correspond to Maurer-Cartan solutions of the algebra of polyvector fields with the nonzero differential [π 1 , •].Using α 0 to convert polyvector fields to differential forms we get the complex in which the bracket with π 1 becomes the de Rham differential.Since the formal de Rham complex is exact in degree two, there is a unique quantization with the h-linear part π 1 .Hence the above isomorphism modulo h extends to an isomorphism σ D .If it can be extended to pair (σ D , σ M ) compatible with the module action, then I h , resp, J is the annihilator of E h,y /h E h,y , resp.M/hM, which implies compatibility with ideals stated in (1).
For existence of σ M in part (2) assume that compatibility with ideals does hold, and first construct the isomorphism modulo h and the lift it inductively modulo higher powers of h.Indeed, using σ M we can view E h,y /h E h,y and M/hM as projective (hence free) modules of the same rank over the local ring O Y,y .Hence there is an isomorphism ρ 0 : M/hM → E h,y /h E h,y To lift it to E h,y and M, take the standard space of generators k ⊕e ⊂ M, and choose any lift ρ : k ⊕e → E h,y of ρ 0 | k ⊕e .Consider the subalgebra D ′ ⊂ D with (topological) generators x 1 , . . ., x n , y q+1 , . . ., y n , h and the map σ ′ : . By a version of Nakayama's Lemma it is an isomorphism of k[[h]]-modules.It induces an isomorphism with (D/D y 1 , . . ., y q ) ⊕e = M (as D-modules) precisely when y s • ρ(v) = 0 for any v ∈ k ⊕e and s ≤ q.Our goal is to adjust ρ to achieve this condition inductively, ensuring that the vanishing holds modulo h l for l ≥ 1.This obviously works for l = 1 as y s acts by zero on E h,y /h E h,y .To make an inductive step, suppose that we have ρ l+1 : k ⊕e → E h,y /h l+1 E h,y and that by inductive assumption y s • ρ l+1 (v) is divisible by h l for all v and s ≤ q.Let U = Im(ρ l+1 ) and let u 1 , . . ., u e ∈ U be the images of the standard basis vectors.By assumption, We are looking for elements where Ham q is the algebra of Hamiltonian derivations of k[[x 1 , . . ., x q , y 1 , . . ., y q ]].The two compositions Der(D, M) → Ham q agree and this gives rise to a short exact sequence: Following the pattern of in section 5 in [Ye], section 6 in [VdB] or sections 2, 3 in [DGW] we see that all pairs (σ D , σ M ) are parameterized by a Harish Chandra torsor P D,M over the pair (Der(D, M), Aut(D, M)).Similarly, all isomorphisms σ E are parameterized by a Harish Chandra torsor P E over the pair (Der(E), Aut(E)).We note here that for P D,M the connection form γ of Section 3.2 is an isomorphism (such torsors are called transitive) while for P E the short exact sequence (8) implies that this torsor is foliated over F .

4.3
The class τ Y is the image of τ Lie .
By Proposition 3.3 and Section 4.0.3 in [BNT] the characteristic class is equal to the image, with respect to the Gelfand-Fuks map of the torsor P D,M , of the class where the factors other than e −c are defined at the end of Section 3.1 and c is obtained from the extension class of 0 by restricting to the subalgebra Der(D) J ⊂ Der(D) and then pulling back under the surjection of (7).For the factors other than e −c we use the fact that the components of τ Y can be defined by using the Chern-Weil construction on the torsor of symplectic frames in Q and the torsor of usual frames in N, E. Since these torsors can be induced from P D,M , we can apply compatibility of Gelfand-Fuks map with induced torsors to reinterpret the classes via Lie algebra cohomology of Der(D, M).We record for future reference that 4.4 Reduction to class τ Dp and end of proof.
Lemma 4.3 The cohomology class τ Lie is represented by a cocycle which vanishes if one of its arguments is in 1 h J .Hence τ Lie is a pullback of a cohomology class of Der(E) via the surjection in (8) and that class is further equal to the class of Lemma 3.4 Proof.
Step 1.We first recall the definitions.Assign the elements in gl e (k) ⊂ Der(D, M) degree 0 and keep assuming that deg h = 2, deg x i = deg y j = 1.Then any element of Der(D, M) is a possibly infinite sum of homogeneous elements of degree ≥ −1 and the Lie bracket is homogeneous.
Then gl e and sp 2p are spanned by elements 1 h x i y j , 1 ≤ i, j ≤ q and 1 h x s x t , 1 h y s y t , 1 h (x s y t + y t x s ), q + 1 ≤ s, t ≤ p + q, respectively, and the degree zero part of Der(D, M) splits as gl e ⊕ gl q ⊕ sp 2p ⊕ W where W is spanned by 1 h y j y t , 1 h x s y j with 1 ≤ j ≤ q, 1 ≤ t ≤ (p + q), (q + 1) ≤ s ≤ (p + q) (we note here that in the specified ranges the variables commute).This gives a projection Der(D, M) → gl e ⊕ gl q ⊕ sp 2p sending the elements of nonzero degree to zero, and vanishing on W .We can combine it with a natural projection to any of the three factors on the right hand side, to be used for calculation of classes ch Lie (gl e ), A Lie (sp 2p ), exp(− c1,Lie(gl q ) 2 ) in the definition of τ Lie .The curvature defined in (4) is not zero only when its arguments have degrees −1, 0 or 1.The same applies to the degree zero component c 0 of c.We recall here that c 0 is computed with respect to the projection onto k which vanishes on elements of non-zero degrees, trace zero matrices in gl e and on the subspaces sp 2p , W , and on the elements of the type 1 h (x i y j + y j x i ).
Step 2. Let us show that the A Lie class is pulled back from the quotient by 1 h J .In fact, consider the curvature C(u ∧ v) ∈ h for the projection onto h = sp 2p and u ∈ 1 h J of degree −1, 0 or 1.It follows from ( 8) that 1 h J is a Lie ideal which has zero projection onto sp 2p , so all positive components of A Lie (sp 2p ) vanish if one of the arguments is in 1 h J .
Step 3. Let us prove that the cochain representing the class ch Lie (gl e )exp(− 1 2 c 1,Lie (gl q ) − c 0 ) is zero if one of its arguments is in 1 h J .Since c 0 vanishes on elements 1 h (x i y j + y j x i ) we have Since b ii is the invariant polynomial corresponding to c 1,Lie (gl q ) we conclude that c 0 + c 1,Lie (gl q ) corresponds to the linear function on h = gl e ⊕ gl q which sends (X 1 , X 2 ) to 1 e tr(X 1 ).Moreover, since ch Lie (gl e ) comes from tr(exp(x)) and tr(exp(x − α • I)) = tr(exp(x))exp(−α), we can rewrite the above class as the image of the invariant series n under the Chern-Weil map.We denote by X = X 1 − 1 e tr(X 1 ) the trace zero part of X 1 and by S l (X) = 1 l! tr(X 1 ) l the degree l component of the invariant power series.Recall that the Chern-Weil class corresponding to S l (X) is obtained by polarization of S l (X): where the sum is over all permutations σ ∈ S 2n that satisfy σ(2i − 1) < σ(2i).So it suffices to show that C = 0 in gl e if C = C(u ∧ v) with u ∈ 1 h J .This holds since 1 h J is a Lie ideal and its projection onto gl q lands into the subspace of scalar matrices which have trivial X part.
Step 4. It remains to show that the class exp(−(c − c 0 )) is in the image of the pullback under the projection Der(D, M) → Der(E).Recall that c was defined in Section 4.3.Let θ be the pullback of a similar class under Der(D, M) → Der(E).Then by the short exact sequence (9), the sum c − θ is zero (the minus sign is in front of θ is due to the fact that M was considered as a right E-module or, equivalently, a left E op -module).We are using the fact that the sum of two extensions descends to the fiber product over Ham p , that the the pullback of the extension class to the extension algebra Der(D, M) must be zero, and that existence of P D,M allows to use the Gelfand-Fuks map associated to this torsor.Hence, on the level of cohomology c = θ and the same holds for each of the coeffiients in the expantion in powers of h, e.g.c 0 = θ 0 .
The fact that the class pulled back from Der(E) is exactly τ Dp follows from the definition of τ Lie and the vanishing proved.
End of proof of Theorem 1.1.By Section 2.2 (application of Riemann Roch Theorm), the class τ (E h ) is the image of τ Y (E) in the cohomology of Y .
By Proposition 3.3 the class τ Y (E) in the de Rham cohomology of Y is the image of a class τ Lie under the Gelfand-Fuks map associated to the Harish-Chandra torsor P D,M of all isomorphisms (σ D , σ M ).
By Lemma 4.3, we can replace the pair (τ Lie , P D,M ) by the pair (τ Dp , P E ) where the torsor P E parameterizes isomorphisms σ E : End O h E h → E (we note here that a choice of (σ D , σ M ) also induces a choice of σ E ).