ATIYAH AND TODD CLASSES ARISING FROM INTEGRABLE DISTRIBUTIONS ZHUO CHEN, MAOSONG XIANG, AND PING XU A

ZHUO CHEN, MAOSONG XIANG, AND PING XU

ABSTRACT. In this paper, we study the Atiyah class and Todd class of the DG manifold(F[1], dF)corre- sponding to an integrable distributionF ⊂ T_KM =T M ⊗_RK, whereK = RorC. We show that these two classes are canonically identical to those of the Lie pair(T_KM, F). As a consequence, the Atiyah class of a complex manifoldX is isomorphic to the Atiyah class of the corresponding DG manifold(T_X^0,1[1],∂).¯ Moreover, ifXis a compact K¨ahler manifold, then the Todd class ofXis also isomorphic to the Todd class of the corresponding DG manifold(T_X^0,1[1],∂).¯

CONTENTS

1. Introduction 1

2. Cohomology of DG manifolds out of integrable distributions 4

2.1. The contraction data 4

2.2. Proof of TheoremA 10

2.3. Application 10

3. Atiyah and Todd classes 13

3.1. Atiyah and Todd classes of DG manifolds 13

3.2. Atiyah and Todd classes of the Lie pair(T_KM, F) 13

3.3. Splittings and connections 14

3.4. Proof of TheoremB 15

References 18

1. INTRODUCTION

Atiyah classes were introduced by Atiyah [1] to characterize the obstruction to the existence of holomorphic connections on a holomorphic vector bundle over a complex manifold. In [5], Kapranov showed that the Atiyah class α_X ∈ H¹(X, T_X^∨ ⊗End(T_X))of a K¨ahler manifoldXinduces an L∞ algebra structure on Ω^0,•−1_X (T_X)(see [7,8] for the complex manifold case), which plays an important role in the formalism of Rozansky-Witten theory. In his seminar work [6], Kontsevich indicated a deep link between the Todd genus of complex manifolds and the Duflo element of Lie algebras. See [9] for the formality theorem for smooth DG manifolds, which implies the Kontsevich-Duflo theorem for Lie algebras and Kontsevich’s theorem for

Research partially supported by NSF grants DMS-1406668 and DMS-1707545.

1

complex manifolds [6] under a unified framework. Inspired by the work of Kapranov and Kontsevich, Chen, Sti´enon and Xu introduced the Atiyah class of Lie algebroid pairs in [3], which encodes both the Atiyah class of complex manifolds and the Molino class [14] of foliations as special cases. Towards a different direction, Mehta, Sti´enon and Xu [13] introduced Atiyah classes and Todd classes of DG manifolds. The Duflo element of a Lie algebragis identified with the Todd class of the associated DG manifold(g[1], dCE).

Let F ⊂ T_KM = T M ⊗_RKbe a subbundle whose space of sections forms an integrable distribution, whereKis either the fieldRorC. WhenK=R, it corresponds to the tangent bundle of a regular foliation F inM. We will also say that F is a regular foliation. Then the graded manifoldF[1] together with the Chevalley-Eilenberg differentiald_F on the Lie algebroidFoverKis a DG manifold. Thus we may consider the Atiyah class and the Todd class [13] of this DG manifold. Meanwhile,(T_KM, F)is a Lie algebroid pair overM. We may also consider the Atiyah class and the Todd class [3] of the Lie pair(T_KM, F).

The main purpose of this paper is to investigate the relation between these two types of Atiyah classes and Todd classes. First of all, we prove the following theorem in Section2regarding the cohomology groups of tensor fields of the DG manifold(F[1], d_F).

Theorem A. For any pair of non-negative integers(n, m), there exists a canonical isomorphism Φⁿ_m : H^•(Tⁿ_m(T(F[1])), L_dF)−^∼=→H^•_CE(F,Tⁿm(T_KM/F))

from the cohomology of(n, m)-tensor fieldsTⁿ_m(T(F[1]))of the DG manifold(F[1], dF)to the Chevelley- Eilenberg cohomology of the F-module Tⁿm(T_KM/F) (see Equation (1.1) for notations). Moreover, the following diagram commutes:

H^•(Tⁿ_m¹₁(T(F[1])), L_dF)⊗_KH^•(Tⁿ_m²₂(T(F[1])), L_dF) ^{⊗ //}

(Φⁿm¹1,Φⁿm²2)

H^•(Tⁿ_m¹₁⁺ⁿ_+m²₂(T(F[1])), L_dF)

Φⁿ_m¹⁺ⁿ²

1+m2

H^•_CE(F,Tⁿm¹1(T_KM/F))⊗_KH^•_CE(F,Tⁿm²2(T_KM/F)) ^{⊗ //}H^•_CE(F,Tⁿm¹1⁺ⁿ+m²2(T_KM/F)).

Moreover, we have a homotopy contraction betweenΓ(M,∧^•F^∨⊗∧^•(T_KM/F))and the space of polyvec- tor fieldsΓ(F[1],∧^•T(F[1]))of the DG manifold(F[1], d_F). Note that

(Γ(F[1],∧^•T(F[1])), L_dF,∧,[−,−]_SN)

is a differential Gerstenhaber algebra, thus also a(−1)-shifted derived Poisson algebra [2], where[−,−]_SN is the Schouten-Nijenhuis bracket. Op cit, a(−1)-shifted derived Poisson algebra structure is constructed on Γ(M,∧^•F^∨ ⊗ ∧^•(T_KM/F)). We show in Proposition2.31that this structure in fact results from the homotopy transfer of the differential Gerstenhaber algebra structure in Γ(F[1],∧^•T(F[1]))via the afore- mentioned contraction data.

Then we prove the following main theorem in Section3.

Theorem B. Under the canonical isomorphism as in TheoremA,

(1) The Atiyah classα_F[1] of the DG manifold(F[1], d_F)corresponds to the Atiyah classα_T

KM/F of the Lie pair(T_KM, F);

(2) The scalar Atiyah classesck(F[1])(k= 1,2,· · ·) of the DG manifold(F[1], dF)correspond to the scalar Atiyah classesc_k(T_KM/F)of the Lie pair(T_KM, F);

(3) The Todd classTd_F[1] of the DG manifold(F[1], d_F)corresponds to the Todd classTd_T

KM/F of the Lie pair(T_KM, F).

As a main application, consider a complex manifold X. Let T_X^1,0 and T_X^0,1 be its holomorphic and anti- holomorphic tangent bundle, respectively. By considering the particular integrable distributionF :=T_X^0,1 ⊂

T_CX, we get a Lie pair (T_CX, T_X^0,1) and a DG manifold (T_X^0,1[1],∂), where¯ ∂¯ : Ω^0,•_X → Ω^0,•+1_X is the Dolbeault differential operator. Applying TheoremsAandB, we have the following

Theorem C. LetΘX andΩX be the holomorphic tangent sheaf and holomorphic cotangent sheaf of X, respectively. Then

(1) There exists a canonical isomorphism

Φⁿ_m: H^•(Tⁿ_m(T(T_X^0,1[1])), L∂¯)−→^∼= H^•(X,(Ω_X)^⊗m⊗O_X (Θ_X)^⊗n),

whereH^•(X,(ΩX)^⊗m⊗_OX(ΘX)^⊗n)denotes the sheaf cohomology of(ΩX)^⊗m⊗_OX (ΘX)^⊗n. (2) The canonical isomorphismΦ¹₂ sends the Atiyah classα_T0,1

X [1] of the DG manifold (T_X^0,1[1],∂)¯ to the Atiyah classαX ∈H¹(X,Ω^⊗2_X ⊗_OXΘX)of the complex manifoldX.

(3) The canonical isomorphismΦ⁰_• sends the Todd classTd_T^0,1

X [1] of the DG manifold (T_X^0,1[1],∂)¯ to the Todd classTd_T1,0

X

∈ ⊕_kH^k,k(X)of the Lie pair(T_CX, T_X^0,1).

Notations:

In this note, the symbol K denotes either the field R or C, and graded means Z-graded. By a graded manifold M overK, we mean a pair (M,O_M), where M is a smooth manifold, andO_M is a sheaf of graded commutative K-algebras overM, locally isomorphic toC^∞(U,K)⊗_KS(V\^∨)for a graded vector spaceV overK, andU ⊂M open. We denote byC^∞(M,K)the space ofK-valued smooth functions on M, i.e., the space of global sections ofO_M. A graded vector bundle is a vector bundle in the category of graded manifolds(cf.[12]).

For any graded vector bundleEover a graded manifoldM, denote by

T^q_p(E) =E^⊗q⊗(E^∨)^⊗p, T^q_p(E) = Γ(M;T^q_p(E)), (1.1) the tensor product ofq-copies ofEandp-copies of its dualE^∨, and the corresponding global section space, respectively. In particular, T¹₀(TM) = Γ(TM) is the space of vector fields ofM, which will also be denoted byX(M).

A DG manifold is a pair(M, Q), whereMis a graded manifold, andQ∈X(M)is a homological vector field onM, i.e., a degree+1derivation ofC^∞(M)such that[Q, Q] = 0. A DG vector bundle is a vector bundle in the category of DG manifolds(cf.[12]).

LetE → Mbe a graded vector bundle and assume thatMis a DG manifold. Then a DG-module structure onΓ(E)overC^∞(M)induces a homological vector field onEsuch thatE → Mis a DG vector bundle. In fact, the category of DG vector bundles and the category of locally free DG-modules are equivalent(cf. [13]).

Let(M, Q)be a DG manifold. Then for anyp, q≥0, the vector bundle T^q_p(TM) = (TM)^⊗q⊗(T^∨M)^⊗p,

together with the Lie derivative LQ along the homological vector field Q, is a DG vector bundle over (M, Q). And(∧^q(TM)⊗ ∧^p(T^∨M), L_Q)is a DG subbundle of(T^qp(TM), L_Q).

Acknowledgements: Xiang is grateful to his advisor Xiaobo Liu for his constant support and encourage- ment, to Pennsylvania State University for its hospitality and to China Scholarship Council for the financial support during his 20-months stay at Penn State. We would like to thank Ruggero Bandiera for telling us reference [10] on the tensor product of homotopy contractions. We also wish to thank Damien Broka, Hsuan-Yi Liao and Mathieu Sti´enon for helpful discussions.

2. COHOMOLOGY OFDG MANIFOLDS OUT OF INTEGRABLE DISTRIBUTIONS

In this section, we prove TheoremAconcerning the cohomology of(n, m)-tensor fields on the DG manifold (F[1], d_F)via constructing a homotopy contraction explicitly.

2.1. The contraction data. LetF ⊂T_KM be an integrable distribution. For simplicity, we will denote the quotient bundleT_KM/F byB from now on. There is a short exact sequence of vector bundles overM:

0 ^//F ^{i //}T_KM ^{prB //}B=T_KM/F ^//0. (2.1)

There exists a flatF-connection∇^BottonB, called Bott representation, which is defined by

∇^Bott_V Z := pr_B[V, j(Z)], ∀V ∈Γ(F), Z ∈Γ(B), wherej:B →T_KM is any splitting of the short exact sequence (2.1).

Meanwhile, there associates toF a DG manifold(F[1], dF), whose space ofK-valued smooth functions is C^∞(F[1],K) = Γ(M;∧^•F^∨) : = Ω^•_F.

Here dF : Ω^•_F → Ω^•+1_F is the Chevalley-Eilenberg differential ofF. Let π : F[1] → M be the bundle projection map. For any vector bundleEoverM, we denote the space of smooth sections of the pull-back bundleπ^∗(E)overF[1]by

Γ(F[1], π^∗(E))∼= Γ(M;∧^•F^∨⊗E) := Ω^•_F(E).

In particular, the covariant derivatived_Bof the Bott connection∇^BottonBgives rise to a cochain complex (Ω^•_F(B), dB), which is called the Chevalley-Eilenberg complex of F with respect to B. And dB is also called theF-module structure onB.

The following lemma appeared without explicit proof in [15]. For completeness, we give a proof here.

Lemma 2.2. For any splittingj : B →T_KM of the short exact sequence(2.1), there exists a contraction data

(X(F[1]), L_dF) (Ω^•_F(B), d_B),

hj

φ ψj

satisfying

φ◦ψj = id : (Ω^•_F(B), dB)→(Ω^•_F(B), dB), (2.3) id−ψ_j◦φ= [L_dF, hj] : (X(F[1]), L_dF)→(X(F[1]), L_dF), (2.4) with side conditions

h_j◦ψ_j = 0, φ◦h_j = 0, h²_j = 0, wheredBis the Chevalley-Eilenberg differential ofF with respect to theF-moduleB.

Proof. Let us choose a splitting of the short exact sequence (2.1), i.e., a pair of bundle mapsj : B→T_KM andpr_F : T_KM →F such thatpr_B◦j= id_B, pr_F◦i= id_F, andid_T

KM =j◦pr_B+i◦pr_F: 0 ^//F ^{i //}T_KM

pr_F

oo

pr_B //B

oo j //0.

By abuse of notations, we will use same notations to denote the corresponding splitting of the short exact sequence:

0 ^//Ω^•_F(F) ^{i //}Ω^•_F(T_KM)

pr_F

oo

pr_B //Ω^•_F(B)

oo j //0.

In the sequel, we considerBas a subbundle ofT_KMtransversal toF via the isomorphismT_KM ∼=F⊕B determined byj. The symbolR denotes the spaceC^∞(M,K) ofK-valued smooth functions onM. The

space of vector fields onF[1]is canonically isomorphic to the graded Lie algebra ofK-linear derivations of Ω^•_F:

X(F[1])∼= Der(Ω^•_F).

First of all, we define the mapsφ, ψ_j andh_j as follows:

• Letφbe theΩ^•_F-linear map

φ: = pr_B◦π_∗ : X(F[1])→Ω^•_F(B), (2.5) whereπ∗ : X(F[1])→ Ω^•_F(T_KM)is the tangent map of the bundle projectionπ : F[1]→M. In fact,π∗(X)∈Ω^•_F(T_KM)∼= Ω^•_F ⊗_RDer(R)is determined by

π∗(X)(f) =X(π^∗f)∈Ω^•_F, ∀f ∈R.

• The map

ψ_j : Ω^•_F(B)→X(F[1]) (2.6) is theΩ^•_F-linear extension of theR-linear map

ψ_j : Γ(B)→Der(Ω^•_F)∼=X(F[1]) specified by the following relations:

ψj(Z)(π^∗f) =j(Z)(f), ψj(Z)(ξ) = pr_F^∨(L_j(Z)ξ)∈Ω¹_F, (2.7) for allZ ∈Γ(B), f ∈Randξ∈Ω¹_F. More precisely, we have

ι_V(ψj(Z)(ξ)) =j(Z)hV, ξi − hpr_F[j(Z), V], ξi, (2.8) for allV ∈Γ(F). It is easy to see that the mapψ_jis well-defined.

• Leth_j : X(F[1])→X(F[1])be theΩ^•_F-linear map of degree(−1)determined by h_j(X) = (−1)^{|X |}ι_pr

F(π∗(X)) ∈Der(Ω^•_F)∼=X(F[1]) (2.9) for allX ∈X(F[1]). Herepr_F(π∗(X))∈Ω^•_F(F), andιis the contraction operator

ια⊗_RVω:=α∧ι_Vω, ∀α, ω∈Ω^•_F, V ∈Γ(F).

In particular, we have, for anyf ∈R,

hj(X)(π^∗f) = 0, hj(X)(dF(π^∗f)) = (−1)^{|X |}pr_F(π∗(X))(f). (2.10) The rest of proof is divided into several steps:

• The mapφis a cochain map.

It suffices to show that for anyX ∈X(F[1]),

φ([d_F,X]) =d_B(φ(X))∈Ω^•_F(B).

We prove, for anyω ∈Γ(B^∨), that the identity

hω, φ([d_F,X])i=hω, d_B(φ(X))i ∈Ω^•_F (2.11) holds. Note that pr^∨_B(ω) ∈ Ω¹(M) is locally of the form P

kg_kdf_k, wheref_k, g_k ∈ R. It thus follows that

Xg_kd_F(π^∗f_k) =i^∨X g_kdf_k

=i^∨(pr^∨_B(ω)) = 0. (2.12) Now

LHS of (2.11)=hpr^∨_B(ω), π∗[dF,X]i=X

gkhdf_k, π∗[dF,X]i=X

gk[dF,X](π^∗fk)

=X

g_kd_F(X(π^∗f_k))−X

(−1)^{|X |}g_kX(d_Fπ^∗f_k) by Equation (2.12)

=d_FX

g_kX(π^∗f_k)

−X

d_F(π^∗g_k)X(π^∗f_k) + (−1)^{|X |}X

X(π^∗g_k)d_F(π^∗f_k)

=d_Fhω, φ(X)i −X

d_F(π^∗g_k)X(π^∗f_k) +X

d_F(π^∗f_k)X(π^∗g_k).

Here in the last equality we have used the following identity hω, φ(X)i=hpr^∨_B(ω), π∗(X)i=X

gkhdf_k, π∗(X)i=X

gkX(π^∗fk).

On the other hand, to compute the right-hand side of Equation (2.11), we first note that the Lie algebroid cohomology differentialdB^∨ corresponding to the dualF-module structure onB^∨can be computed by the following commutative diagram

Γ(B^∨) ^{dB∨ //}

pr^∨_B

Ω¹_F(B^∨)⊂Ω¹_F(T_KM)

Ω¹(M) ^{dDR //}Ω²(M).

RF

OO

Here the right vertical mapRF is defined by

R_F(α∧β) = (i^∨⊗id)(α⊗β−β⊗α) =i^∨(α)⊗β−i^∨(β)⊗α, for allα, β ∈ Ω¹(M). And the fact thatIm(R_F ◦d_DR◦pr^∨_B) = Ω¹_F(B^∨) ⊂ Ω¹_F(T^∨

KM)follows from the assumption thatF ⊂T_KMis integrable. Thus,

RHS of (2.11)=dFhω, φ(X)i − hd_B^∨(ω), φ(X)i

=dFhω, φ(X)i − hR_FX

dgk∧dfk

,pr_B(π∗(X))i

=d_Fhω, φ(X)i −D

pr^∨_BX

d_F(π^∗g_k)⊗df_k−d_F(π^∗f_k)⊗dg_k

, π∗(X)E

=dFhω, φ(X)i −X

dF(π^∗gk)X(π^∗fk) +X

dF(π^∗fk)X(π^∗gk).

• The mapψ_jis a cochain map, i.e.,ψ_j ◦d_B=L_dF ◦ψ_j. We have

ψj(dB(α⊗_RZ)) =dF(α)⊗_Rψj(Z) + (−1)^|α|α⊗_Rψj(dB(Z)) and

L_dF(ψ_j(α⊗_RZ)) =d_F(α)⊗_Rψ_j(Z) + (−1)^|α|α⊗_RL_dF(ψ_j(Z)), for anyα⊗_RZ ∈Ω^•_F(B), whereα∈Ω^•_F, Z ∈Γ(B). Thus it suffices to prove that

ψj(dB(Z)) =LdF(ψj(Z)) = [dF, ψj(Z)]∈X(F[1]).

Let us examine this relation on the generators{π^∗f, dF(π^∗f) : f ∈R}of theK-algebraΩ^•_F: On the one hand, for anyV ∈Γ(F), by the defining Equation (2.7) ofψ_j, we have

ι_V(ψ_j(d_B(Z))(π^∗f)) =ψ_j(∇^Bott_V (Z))(π^∗f) =j(pr_B([V, j(Z)]))(f)

= [V, j(Z)](f)−pr_F([V, j(Z)])(f)

=V(j(Z)(f))−(j(Z)(V(f))−pr_F([j(Z), V])(f)) by Equation (2.8)

=ιV([dF, ψj(Z)](π^∗f)).

Therefore, we have

ψ_j(d_B(Z))(π^∗f) = [d_F, ψ_j(Z)](π^∗f). (2.13) On the other hand, we have

[d_F, ψ_j(Z)](d_F(π^∗f)) =−d_F([d_F, ψ_j(Z)](π^∗f)) by Equation(2.13)

=−d_F(ψ_j(d_B(Z))(π^∗f))

=−[d_F, ψj(dB(Z))](π^∗f) +ψj(dB(Z))(dF(π^∗f)) by Equation(2.13)

=−ψ_j(d²_B(Z))(π^∗f) +ψ_j(d_B(Z))(d_F(π^∗f))

=ψ_j(d_B(Z))(d_F(π^∗f)).

Hence,ψ_jis a cochain map.

• Proof of Equation(2.3). We first claim that

π∗◦ψj =j: Ω^•_F(B)→Ω^•_F(T_KM), (2.14) Since bothψj andπ∗areΩ^•_F-linear, it suffices to show that

π∗(ψj(Z)) =j(Z)∈Γ(T_KM)∼= Der(R), ∀Z ∈Γ(B).

In fact, by the defining formula (2.7) ofψj, we have

π∗(ψj(Z))(f) =ψj(Z)(π^∗f) =j(Z)(f), ∀f ∈R.

This proves Equation (2.14). Hence,

φ◦ψ_j = pr_B◦π_∗◦ψ_j = pr_B◦j= id.

• Proof of Equation(2.4). We prove that

X −ψ_j(φ(X)) = [L_dF, h_j](X) = [d_F, h_j(X)] +h_j([d_F,X]),

for allX ∈X(F[1]). It suffices to show that both sides coincide when acting on elements{π^∗f, d_F(π^∗f) : f ∈R}. That is, for anyf ∈R,

(X −ψj(Φ(X)))(π^∗f) = [dF, hj(X)](π^∗f) +hj([dF,X])(π^∗f) = [dF, hj(X)](π^∗f), (2.15) (X −ψ_j(Φ(X)))(d_F(π^∗f)) = [d_F, h_j(X)](d_F(π^∗f)) +h_j([d_F,X])(d_F(π^∗f)). (2.16)

In fact, Equation (2.15) can be verified directly:

[dF, hj(X)](π^∗f) = (−1)^{|X |}hj(X)(dF(π^∗f)) by Equation (2.10)

= pr_F(π∗(X))(f) = (π∗(X)−pr_B(π∗(X)))(f) by Equation (2.5)

= (π∗(X)−(φ(X)))(f)

=X(f)−ψ_j(φ(X))(f).

As for Equation (2.16), we compute (X −ψ_j(φ(X)))(d_F(π^∗f))

= (−1)^{|X |}(d_F((X −ψj(φ(X)))(π^∗f))−[d_F,X −ψj(φ(X))](π^∗f)) by Equation (2.15)

= (−1)^{|X |}(dF([dF, hj(X)](π^∗f))−[dF,X](π^∗f) + [dF, ψj(φ(X))](π^∗f))

= (−1)^{|X |}(d_F([d_F, h_j(X)](π^∗f))−[d_F,X](π^∗f) +ψ_j(φ([d_F,X]))(π^∗f))

= [d_F, hj(X)](d_F(π^∗f))−(−1)^{|X |}(π∗([d_F,X])(f)−j(pr_B(π∗([d_F,X])))(f))

= [dF, hj(X)](dF(π^∗f)) + (−1)^{|X |+1}pr_F(π∗([dF,X]))(f) by Equation (2.10)

= [dF, hj(X)](dF(π^∗f)) +hj([dF,X])(dF(π^∗f)).

• Side conditions. Note that by Equation (2.14), (hj◦ψj)(Z) =ι_pr

F(π∗(ψj(Z)))=ι_pr

F(j(Z)) = 0, ∀Z ∈Γ(B).

Since bothh_jandψ_jareΩ^•_F-linear, it follows thath_j◦ψ_j = 0. The verifications ofφ◦h_j = 0and h²_j = 0are similar.

This completes the proof.

We generalize this result to the space of tensor fields onF[1]as follows:

Proposition 2.17. For any splitting j : B → T_KM of the short exact sequence (2.1), there exists a contraction data

(Tⁿ_m(T(F[1])), L_dF) (Ω^•_F(Tⁿm(B)), d_B),

H_mⁿ

Φⁿ_m Ψⁿ_m

satisfying

Φⁿ_m◦Ψⁿ_m = id, Ψⁿ_m◦Φⁿ_m+ [L_dF, H_mⁿ] = id, with side conditions

H_mⁿ ◦Ψⁿ_m= 0, Φⁿ_m◦H_mⁿ = 0, H_mⁿ ◦H_mⁿ = 0.

Here for simplicity we denote bydBthe Chevalley-Eilenberg differential ofF with respect to theF-module Tⁿm(B)for any pair of non-negative integers(m, n).

Proof. Note that all mapsψj, φ, hj of the contraction data areΩ^•_F-linear. TheirΩ^•_F-dual mapsψ_j^∨, φ^∨, h^∨_j give rise to a contraction as well:

(Ω¹(F[1]), L_dF) (Ω^•_F(B^∨), d_B).

h^∨_j

ψ^∨_j φ^∨

By considering theΩ^•_F-dual form of Equations (2.3) and (2.4), it only suffices to show that [LdF, hj]^∨ = [LdF, h^∨_j] : Ω¹(F[1])→Ω¹(F[1]).

In fact, for allα∈Ω¹(F[1]),X ∈X(F[1]), we have

h[L_dF, h^∨_j]α,X i=hL_dF(h^∨_j(α)),X i+hh^∨_j(L_dF(α)),X i

=LdFhh^∨_j(α),X i −(−1)^|α|−1hh^∨_j(α), LdFX i+ (−1)^|α|+1hL_dFα, hj(X)i

= (−1)^|α|(L_dFhα, h_j(X)i − hL_dFα, h_j(X)i) +hα, h_j(L_dFX)i

=hα, L_dF(h_j(X)) +h_j(L_dFX)i=hα,[L_dF, h_j]X i

=h[L_dF, hj]^∨(α),X i.

Define

Ψⁿ_m : = (φ^∨)^⊗m⊗(ψj)^⊗n: (Ω^•_F(Tⁿm(B)), dB)→(Tⁿ_m(T(F[1])), Ld_F), (2.18) Φⁿ_m : = (ψ^∨_j)^⊗m⊗(φ)^⊗n: (Tⁿ_m(T(F[1])), LdF)→(Ω^•_F(Tⁿ_m(B)), dB), (2.19) H_mⁿ =

m

X

i=1

(φ^∨◦ψ^∨_j)^⊗(i−1)⊗h^∨_j ⊗(id)^⊗(n+m−i)+

n

X

l=1

(φ^∨◦ψ^∨_j)^⊗m⊗(ψj◦φ)^⊗(l−1)⊗hj⊗id^⊗(n−l). (2.20) Then according to Lemma2.2, and the tensor trick in [10, Example 2.6], the mapsΨⁿ_m,Φⁿ_m, H_mⁿ constructed

above satisfy the contraction properties.

Moreover, the quasi-isomorphismsΨⁿ_mandΦⁿ_mare, in fact, canonical up to homotopy:

Proposition 2.21. Assume thatbj: B →T_KMis another splitting of the short exact sequence(2.1). Let (Tⁿ_m(T(F[1])), L_dF) (Ω^•_F(Tⁿm(B)), d_B)

Hbⁿ_m

Φbⁿ_m

Ψbⁿ_m

be the associated contraction. Then there exist twoΩ^•_F-linear maps

Θⁿ_m: Ω^•_F(Tⁿm(B))→Tⁿ_m(T(F[1])), Ξⁿ_m:Tⁿ_m(T(F[1]))→Ω^•_F(Tⁿm(B)) such that

Ψbⁿ_m−Ψⁿ_m= Θⁿ_m◦d_B+L_dF ◦Θⁿ_m: Ω^•_F(Tⁿ_m(B))→Tⁿ_m(T(F[1])), (2.22) Φbⁿ_m−Φⁿ_m= Ξⁿ_m◦L_dF +d_B◦Ξⁿ_m : Tⁿ_m(T(F[1]))→Ω^•_F(Tⁿm(B)). (2.23)

According to the defining formulas (2.18) and (2.19) ofΨⁿ_m andΦⁿ_m in the proof of Proposition 2.17, the key step is to study how the quasi-isomorphismψjdepends on the splittingj:

Lemma 2.24. Under the same assumptions as in Proposition 2.21, there exists a Ω^•_F-linear homotopy betweenΨb¹₀ =ψ

bj andψ_j

Θ : Ω^•_F(B)→X(F[1]), i.e.,

ψbj−ψ_j = Θ◦d_B+L_dF ◦Θ : Ω^•_F(B)→X(F[1]). (2.25) Proof. Letθ=bj−j∈Γ(M; Hom(B, F)). Define a map

Θ : Ω^•_F(B)→Der(Ω^•_F)∼=X(F[1]) by

Θ(α⊗_RZ) = (−1)^|α|α⊗ι_θ(Z), (2.26) whereα∈Ω^•_F, Z ∈Γ(B).

We show that thisΘsatisfies Equation (2.25). In fact, both sides of (2.25) areΩ^•_F-linear, since L_dF(Θ(α⊗_RZ)) = (−1)^|α|[dF, α⊗_RΘ(Z)] = (−1)^|α|dF(α)⊗_RΘ(Z) +α⊗_RL_dF(Θ(Z)),

Θ(d_B(α⊗_RZ)) = Θ(d_F(α)⊗_RZ+ (−1)^|α|α⊗_Rd_B(Z))

= (−1)^|α|+1d_F(α)⊗_RΘ(Z) +α⊗_RΘ(d_B(Z)), for allα⊗_RZ ∈Ω^•_F(B), whereα∈Ω^•_F, Z ∈Γ(B).

It thus suffices to show that

ψbj(Z)−ψj(Z) = Θ(dB(Z)) +L_dF(Θ(Z))∈X(F[1]). (2.27) Let us denote by pr_F,prb_F : T_KM → F the projections corresponding to the two splittings j andbj, respectively. Then

prb_F = pr_F−θ◦pr_B : Ω^•_F(T_KM)→Ω^•_F(F).

For anyf ∈R,

(Θ(d_B(Z)) +L_dF(Θ(Z)))(π^∗f) = [d_F,Θ(Z)](π^∗f) = [d_F, ι_θ(Z)](π^∗f) =θ(Z)(π^∗f)

= (ψ_bj(Z)−ψj(Z))(π^∗f);

For anyξ∈Ω¹_F, V ∈Γ(F), we have ι_V(ψ

bj(Z)(ξ)−ψj(Z)(ξ)) by Equation (2.8)

=θ(Z)hV, ξi − hprb_F([bj(Z), V])−pr_F([j(Z), V]), ξi

=θ(Z)hV, ξi − hpr_F([bj(Z), V])−θ(pr_B[bj(Z), V])−pr_F([j(Z), V]), ξi

=θ(Z)hV, ξi − h[θ(Z), V], ξi − hθ(∇^Bott_V Z), ξi

=hV, L_θ(Z)(ξ)i − hθ(ι_VdB(Z)), ξi=ιV(L_θ(Z)(ξ)) +ιV(Θ(dB(Z))(ξ))

=ιV(Ld_F(Θ(Z))(ξ) + Θ(dB(Z))(ξ)).

SinceΩ^•_F is generated byRandΩ¹_F, it follows that Equation (2.27) holds.

Proof of Proposition2.21. Define Θⁿ_m =

n

X

l=1

(φ^∨)^⊗m⊗(ψ

bj)^⊗(l−1)⊗Θ⊗(ψ_j)^⊗(n−l): Ω^•_F(Tⁿm(B))→Tⁿ_m(T(F[1])),

Ξⁿ_m =

m

X

k=1

(ψ^∨

bj)^⊗(k−1)⊗Θ^∨⊗(ψ^∨_j)^⊗(m−k)⊗(φ)^⊗n: Tⁿ_m(T(F[1]))→Ω^•_F(Tⁿ_m(B)).

HereΘ^∨denotes theΩ^•_F-dual ofΘdefined in Equation (2.26).

Then by Lemma2.24, Equations (2.22) and (2.23) follow from straightforward computations.

2.2. Proof of TheoremA. Now we are ready to prove TheoremA:

Proof of TheoremA. Letj : B →T_KM be a splitting of the short exact sequence (2.1). Applying Propo- sition2.17, we obtain an isomorphism

Φⁿ_m: H^•(Tⁿ_m(T(F[1])), L_dF)−^∼=→H^•_CE(F,Tⁿm(B)).

According to Proposition 2.21,Φⁿ_m does not depend on the splittingj, and thus is canonical. Finally, the commutative diagram in TheoremAfollows from the commutative diagram on the cochain level

Tⁿ_m¹

1(T(F[1]))⊗Tⁿ_m²

2(T(F[1])) ^{⊗ //}

Φⁿm¹1⊗Φⁿm²2

Tⁿ_m¹⁺ⁿ²

1+m2(T(F[1]))

Φⁿ_m¹⁺ⁿ²

1+m2

Ω^•_F(Tⁿm¹1(B))⊗Ω^•_F(Tⁿm²2(B)) ^{⊗ //}Ω^•_F(Tⁿm¹1⁺ⁿ+m²2(B)).

This completes the proof.

In the sequel, the contraction data(Φⁿ_m,Ψⁿ_m, H_mⁿ)in Proposition2.17will simply be denoted by(Φ,Ψ, H):

(Tⁿ_m(T(F[1])), L_dF) (Ω^•_F(Tⁿ_m(B)), d_B).

H

Φ Ψ

It is clear that(Φ,Ψ, H)induces a contraction data on the subcomplexes as well

(Γ(F[1],∧ⁿT(F[1])⊗ ∧^mT^∨(F[1])), LdF) (Ω^•_F(∧ⁿB⊗ ∧^mB^∨), dB).

H

Φ Ψ

(2.28) The following result is an immediate consequence, which implies that cohomologies ofΓ(F[1],∧ⁿT(F[1])⊗

∧^mT^∨(F[1]))are bounded from above:

Corollary 2.29. The cohomology

H^r(Γ(F[1],∧ⁿT(F[1])⊗ ∧^mT^∨(F[1])), L_dF) vanishes under any of the following conditions:

(1) m >rank(B);

(2) n >rank(B);

(3) r >rank(F).

2.3. Application. Consider the casem= 0. The contraction data reduces to (Γ(F[1],∧ⁿT(F[1])), L_dF) (Ω^•_F(∧ⁿB), d_B),

H

Φ Ψ

(2.30) where

Ψ =∧ⁿψ_j : Ω^•_F(∧ⁿB)→Γ(F[1],∧ⁿT(F[1])), Φ =∧ⁿφ: Γ(F[1],∧ⁿT(F[1]))→Ω^•_F(∧ⁿB),

H=

n

X

l=1

∧^(l−1)(ψ_j◦φ)∧h_j∧(id)^∧(n−l) : Γ(F[1],∧ⁿT(F[1]))→Γ(F[1],∧ⁿT(F[1])).

Note that bothΩ^•_F(∧^•B)andΓ(F[1],∧^•T(F[1])), together with the obvious wedge product, are commuta- tive graded algebras. It is clear that bothΨandΦare compatible with the wedge product, thus are morphisms of commutative algebras. However,Hin general is not a derivation. In fact, we have

H(X₁∧ X₂) =H(X₁)∧ X₂+ (Ψ◦Φ)(X₁)∧H(X₂), ∀X_i ∈Γ(∧^•T(F[1])).

Note that(Γ(F[1],∧^•T(F[1])), L_dF)together with the wedge product and the Schouten-Nijenhuis bracket [−,−]_SN, forms a differential Gerstenhaber algebra. By the standard homotopy transfer technique, we recover the following

Proposition 2.31. Given a splittingj: B →T_KM of the short exact sequence(2.1), then

(1) there exists a(−1)-shifted derived Poisson algebra structure{λ_k}_k≥1onΩ^•_F(∧^•B)such thatλ₁ = dB.

(2) this (−1)-shifted derived Poisson algebra structure coincides with the one out of the Lie pair (T_KM, F)as in[2, Proposition 4.3].

The following technical lemma is needed:

Lemma 2.32. For anyX₁,X₂∈X(F[1])andZ ∈Γ(B), we have

φ([hj(X₁), hj(X₂)]) = 0, (2.33) hj([hj(X₁), ψj(Z)]) = 0. (2.34) Proof. Note that the space of vertical vector fields

X(F[1])^vertical={ι_V ∈Der(Ω^•_F) : V ∈Ω^•_F(F)} ∼= ker(π∗) is a Lie subalgebra ofX(F[1]). SinceIm(h_j)⊂X(F[1])^vertical, it follows that

φ([hj(X₁), hj(X₂)]) = pr_B(π∗([hj(X₁), hj(X₂)])) = 0.

Meanwhile, Since

π∗([h_j(X₁), ψ_j(Z)])(f) = [h_j(X₁), ψ_j(Z)](π^∗f) =h_j(X₁)(j(Z)(f)) = 0,

for all f ∈ R, it follows that π∗([hj(X₁), ψj(Z)]) = 0. According to Equation (2.9), the latter implies

Equation (2.34), as desired.

Proof of Proposition2.31. We first use the contraction data (2.30) to prove(1): Note that bothΩ^•_F(∧^•B)and Γ(F[1],∧^•T(F[1]))carry natural graded commutative algebra structures, i.e., the wedge products. More- over,

(Γ(F[1],∧^•T(F[1])), L_dF,∧,[−,−]_SN)

is a differential Gerstenhaber algebra, and therefore is also a(−1)-shifted derived Poisson algebra. Since bothΨandΦare morphisms of the underlying commutative algebras, by the homotopy transfer for derived Poisson algebras [2, Theorem 2.16], there induces a(−1)-shifted derived Poisson algebra structure{λ_k}_k≥1 onΩ^•_F(∧^•B), whose first bracketλ1isdBand higher brackets are defined iteratively op cit.

To prove(2), let us work out higher brackets{λ_k}_k≥2explicitly:

k= 2. The binary bracketλ₂is, by definition

λ₂(Z₁,Z₂) = Φ([Ψ(Z₁),Ψ(Z₂)]_SN), ∀Z₁,Z₂∈Ω^•_F(∧^•B).

More precisely, we consider the situation on generating elements:

λ₂(Z₁, Z₂) =φ([ψ_j(Z₁), ψ_j(Z₂)]_SN) = pr_B([j(Z₁), j(Z₂)]), ∀Z_i∈Γ(B), i= 1,2, λ2(Z1, ξ) =φ([ψj(Z1), ψj(ξ)]SN) =ψj(Z1)(ξ) = pr_F^∨(L_j(Z1)ξ), ∀Z₁ ∈Γ(B), ξ∈Ω¹_F. Similarly, it follows that

λ2(Z1, f) =j(Z1)(f), λ2(ω1, ω2) = 0, for allf ∈R=C^∞(M,K)andωi∈Ω^•_F.

k= 3. The trinary bracketλ3is given by λ3(Z₁,Z₂,Z₃) = X

σ∈sh(2,1)

±Φ([H([Ψ(Z_σ(1)),Ψ(Z_σ(2))]SN),Ψ(Z_σ(3))]SN),

for allZ_i∈Ω^•_F(∧^•B),1≤i≤3, wheresh(2,1)is the set of(2,1)-shuffles, and±is some proper Koszul sign. In particular, we have, on the generating elements,

λ3(Z1, Z2, ξ) =φ([hj([ψj(Z1), ψj(Z2)]SN), ψj(ξ)]SN) =ι_{prF[j(Z1),j(Z2)]}(ξ),

for allZ1, Z2 ∈Γ(B)andξ ∈Γ(F^∨), wherepr_F : Γ(T_KM) →Γ(F)is the projection. Further- more, since

h_j([X, ψ_j(ξ)]_SN) =h_j(X(ξ)) = 0, ∀X ∈X(F[1]), ξ ∈Γ(F^∨)⊂Ω^•_F, (2.35) it follows that λ₃ vanishes if restricted to Γ(B) ×Γ(B) ×Γ(B),Γ(B)×Γ(F^∨)×Γ(F^∨) and Γ(F^∨)×Γ(F^∨)×Γ(F^∨).

k= 4. The 4th bracketλ₄acting on the generating elements is of form λ₄(Z₁, Z₂, Z₃, Z₄) = X

σ∈sh(2,2)

±φ([h_j([ψ_j(Z_σ(1)), ψ_j(Z_σ(2))]_SN), h_j([ψ_j(Z_σ(1)), ψ_j(Z_σ(2))])]_SN)

+ X

τ∈sh(2,1,1)

±φ([h_j([hj([ψj(Z_τ(1)), ψj(Z_τ(2))]SN), ψj(Z_τ(3))]SN), ψj(Z_τ(4))]SN),

for allZi ∈Γ(B),1≤i≤4. It follows from Lemma2.32thatλ4(Z1, Z2, Z3, Z4) = 0. Moreover, it follows from Equations (2.34) and (2.35) that λ4 also vanishes with one or more inputs from Γ(F^∨).

k≥5. TheL∞-brackets{λ_k}_k≥5vanish for similar reasons.

It thus follows that the generating relations of the(−1)-shifted derived Poisson algebra structure onΩ^•_F(∧^•B), are exactly those of [2, Proposition 4.3]. This concludes the proof.

Remark2.36. The(−1)-shifted derived Poisson algebra structure onΩ^•_F(∧^•B)induces a strong homotopy Lie-Rinehart algebra structure on the subspaceΩ^•_F(B). This result is due to Vitagliano [15].

According to Theorem 1.1 in [2], this (−1)-shifted derived Poisson algebra is in fact canonical up to iso- morphism. On the other hand,(H^•(Γ(F[1],∧^•T(F[1])), L_dF),∧,[−,−]_SN)is canonically a Gerstenhaber algebra. As an immediate consequence of Proposition2.31and TheoremA, we have

Corollary 2.37. There is a canonical isomorphism of Gerstenhaber algebras

Ψ : (H^•_CE(F,∧^•B),∧, λ₂)−→^∼= (H^•(Γ(F[1],∧^•T(F[1])), L_dF),∧,[−,−]_SN).

3. ATIYAH ANDTODD CLASSES

3.1. Atiyah and Todd classes of DG manifolds. Let(M, Q)be a DG manifold. The Atiyah classαM ∈ H¹(T¹₂(TM), L_Q)of(M, Q)is, by definition, the Atiyah class of the tangent DG vector bundleTM → M with respect to the DG Lie algebroidTM[13]: Choose aTM-connection∇on the tangent bundleTM.

Then there associates a degree+1LQ-cocycleα^∇_M∈T¹₂(TM)defined by

α^∇_M(X, Y) =LQ(∇_XY)− ∇_LQ(X)Y −(−1)^|X|∇_XLQ(Y), ∀X, Y ∈Γ(TM), (3.1) which will be called the Atiyah cocycle ofMwith respect to∇. The cohomology class

αM = [α^∇_M]∈H¹(T¹₂(TM), L_Q),

which does not depend on the choice of∇, is called the Atiyah class of the DG manifold(M, Q).

Note that

αM ∈H¹(T¹₂(TM), L_Q)∼= H¹(Ω¹(M)⊗_C^∞_(M,K)Γ(End(TM)), L_Q).

For any positive integerk, one can formα^k_M, the image ofα^⊗k_M under the natural map

⊗^k

KH¹(Ω¹(M)⊗_C∞(M,K)Γ(End(TM)), L_Q)→H^k(Ω^k(M)⊗_C∞(M,K)Γ(End(TM)), L_Q) induced by the wedge product inΩ^•(M)and the composition inEnd(TM).

The scalar Atiyah classes [13] of the DG manifold(M, Q)are defined by c_k(M) : = 1

k!

i 2π

k

str(α^k_M)∈H^k(Ω^k(M), L_Q), k= 1,2,· · ·, (3.2) wherestr : Γ(End(TM))→C^∞(M,K)is the supertrace map.

The Todd class [13] of the DG manifold(M, Q)is defined by TdM : = Ber

αM

1−e^−αM

∈Y

k≥0

H^k(Ω^k(M), L_Q), (3.3) whereBer : Γ(End(TM))→C^∞(M,K)is the Berezinian (or the superdeterminant) map [11].

In this paper, we focus on the DG manifold (F[1], dF) corresponding to an integrable distribution F ⊂ T_KM. The Atiyah, scalar Atiyah and Todd classes of(F[1], d_F), are denoted respectively by:

α_F[1] ∈H¹(T¹₂(T(F[1])), LdF),

ck(F[1]) ∈H^k(Ω^k(F[1]), LdF), k= 1,2,· · · Td_F[1] ∈Y

k≥0

H^k(Ω^k(F[1]), LdF).

3.2. Atiyah and Todd classes of the Lie pair (T_KM, F). Given an integrable distribution F ⊂ T_KM, (T_KM, F)is aK-Lie algebroid pair (or Lie pair for short). We briefly recall from [3] the Atiyah and Todd classes of the Lie pair(T_KM, F).

Recall that there is a short exact sequence of vector bundles overM:

0 ^//F ^//T_KM ^//T_KM/F ^//0.

Let us choose a splittingj:T_KM/F →T_KMof the above short exact sequence and aT_KM-connection∇ onT_KM/F extending the BottF-connection. The associated Atiyah cocycle

R^∇∈Ω¹_F(T¹2(T_KM/F)) = Γ(M;F^∨⊗F^⊥⊗End(T_KM/F)),