Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids

In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Transform. Groups 13 (2008), 727-755], and with a connectivity assumption which we here show to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681-721] and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121-169].


Introduction
L a g(s, s ′ ) = g(∇ bas a s, s ′ ) + g(s, ∇ bas a s ′ ), s, s ′ ∈ Γ(Ad(A)). For example, the familiar statement that there exists a Riemannian (i.e., a torsion-free and metric) connection associated with a Riemannian metric on M implies that, for tangent bundles A = T M , these characteristic classes char(A) vanish.
In degree one, char 1 (A) recovers the modular class of A [8], the obstruction to the existence of an invariant transverse measure, first discovered in the context of Poisson manifolds [22,29] as the 'Poisson analogue of the modular automorphism group of a von Neumann algebra'. There is an extensive literature about this important class (see the survey [20]), which is arguably the only reasonably well-understood among the intrinsic ones. It has been generalized to various geometric contexts [3,16,17,18,21,26,27,28], and play a fundamental role in many constructions [3,7,8,10,11,15,24,25,30].
The purpose of this short note is to show that intrinsic characteristic classes are invariant under the following version of weak Morita equivalence [ Versions of this result have appeared in the literature in various forms; we here quote those most pertinent to our setting.
In [13,Theorem 4.2] it was shown, building on previous work [12], that the modular class is a Morita invariant for locally unimodular Poisson manifolds. Shortly afterwards, secondary and intrinsic characteristic classes were introduced (see [4,5,6,9,23]), and in [5,Corollary 8] it was proved that the intrinsic characteristic classes of Poisson manifolds of degree (2q−1) are invariant under Morita equivalences whose fibres are at least homologically (2q − 1)-connected; it is later extended to weak Morita equivalences of Lie algebroids under a similar connectivity condition [14,Example 6.16].
More recently, it was proved in [21,Theorem 3.10] that the modular class is functorial with respect to arbitrary transverse maps -thus dropping the connectivity condition -and they inquire authors pose the question in [21, (iii), p. 729] about the behavior of higher intrinsic characteristic classes under morphisms. It was this question that piqued our interest, and which our Main Theorem seeks to answer. The paper is organized as follows: our conventions are discussed in Section 2, where we summarize the construction of primary-, secondary-and intrinsic characteristic classes of Lie algebroids from [4,6], referring there to proofs. In Section 3 we prove our Main Theorem: as we explain there, this result is a straightforward consequence of the case of pulling back a Lie algebroid A on M by a submersion p : Σ → M , and our proof in that case reduces to the construction of appropriate connection and metric on Ad(p ! (A)), so that the adjoint connection of p ! (A) splits as a direct sum of the pullback of the adjoint connection of A and a metric subconnection.

Characteristic classes
In this section, we give a summary of the main results and constructions needed to contextualize our discussion, referring to the appropriate references for further details. §1. For vector bundle E and D on M , we denote by Ω p nl (E; D) the space of nonlinear forms of degree p on E with values in D -that is, the linear subspace of Hom(∧ p Γ(E), Γ(D)) consisting of those elements ω which decrease support, in the sense that ω(e 1 , ..., e p ) is identically zero around any point around which some e i ∈ Γ(E) vanishes identically. When D is the trivial line bundle, we write Ω p nl (E), and we note that Ω p nl (E; D) is a module over Ω p nl (E). Linear forms ω ∈ Ω p (E, D) = Γ(∧ p E * ⊗ D) are identified with those elements of Ω p nl (E; D) which are C ∞ (M )-linear in their entries. There are obvious variations for linear forms, and when D is complex or graded; see [1,4]. • a dual nonlinear connection ∇ ∨ of A on D * , defined by the condition that Note that every Hermitian metric h is invariant under some nonlinear connection; e.g., ∇ m : is another subconnection, we say that ∇ splits as a direct sum, and write and it is always the case that d ∇ R ∇ = 0. If R ∇ = 0, we call ∇ a nonlinear representation. Because the supertrace Tr s (T ) = Tr(T 00 ) − Tr(T 11 ) induces a linear map intertwining derivations, it follows in general that Tr s (R q ∇ ) ∈ Ω 2q nl (A; C) are d A -closed for every integer q; see [4].
for all q (see [6]). A nonlinear connection ∇ of A on D will be called a connection up to homotopy if it is equivalent to a connection; in this case, we will write ∇ : A D. Both connections and connections up to homotopy are preserved by all operations on nonlinear connections described in §3.- §7. Note that,for a connection up to homotopy ∇, Tr s (R q ∇ ) are linear forms, Tr s (R q ∇ ) ∈ Ω 2q (A). §9. A representation up to homotopy 2 is a connection up to homotopy for which R ∇ vanishes identically.
In that case, d ∇ turns Ω In the remainder of this section, we recall the discussion in [6], referring there to proofs and further details.
Lemma. There is a rule cs which assigns to all non-negative integers p, q 0 and connections up to homotopy ∇ 0 , ..., ∇ p : A D, a cochain with the property that, for every permutation σ and Hermitian metric h on D: 1 As explained in [21], it is best to think that a connection ∇ : 2 For the convenience of the reader, we chose to maintain the term representation up to homotopy as it appears in [4,6], in spite of the fact that that terminology has come to mean something else [1].
Such cochains are given explicitly by where ⌊t⌋ the greatest integer no greater than t and:  . Then for ω ∈ Ω p+q (pr ! (A)) and sections a 1 , ..., aq ∈ Γ(A), define − ∆ p ω so that the identity below is satisfied: Definition. The intrinsic characteristic classes char q (A) ∈ H 2q−1 (A) of the Lie algebroid A are the secondary characteristic classes u q (∇ ad ) of the adjoint representation up to homotopy ∇ ad .
Note that it follows from the discussion in the Main Example, and item b) of Proposition 2, that char(A) can be alternatively defined as the unique element H odd (A) which pulls back under the Lie algebroid map (pr, id) : J 1 (A) → A to the secondary characteristic class u(∇ j1 ) of the canonical representation of J 1 (A) on Ad(A).
Example. The modular class of A coincides with mod(A) = 2π char 1 (A) ∈ H 1 (A).

Proof of the Main Theorem
While primary and secondary characteristic classes are functorial with respect to pullbacks essentially by inspection of the construction, for intrinsic characteristic classes the situation is slightly more intricate, because the adjoint representation up to homotopy of a pullback is not itself a pullback representation up to homotopy. The following special case will turn out to be key: Proof of Proposition 4. Let i : X ֒→ M be a closed embedding transverse to A, and p : N X := T M | X /T X → X the normal bundle to X. By the normal form theorem in [2], we can find an open subset U ⊂ N X, and an isomorphism of Lie algebroids (Φ, φ) : , such that the following triangle of morphisms of Lie algebroids commutes is the embedding of N as the graph of φ. Because pr 2 is a surjective submersion, φ is transverse to A exactly when i is transverse to pr ! 2 (A). Hence To do so, it is enough to give a recipe which to a connection ∇ : T M A and metrics g A on A and g M on T M , assigns a connection ∇ : T Σ p ! (A), and metrics g p ! (A) on p ! (A) and g Σ on T Σ, such that ( * * ) cs(∇ bas , ∇ bas,g ) = p * cs(∇ bas , ∇ bas,g ), where g = (g A , g M ) and g = (g p ! (A) , g Σ ). Our recipe for (∇, g p ! (A) , g Σ ) will depend on choices of a metric g V on the vertical bundle V = ker p * , and an Ehresmann connection H ⊂ T Σ for p, all of which we fix once and for all. Denote by h : p * (T M ) → T Σ the horizontal lift associated with H and by V the subbundle V ⊕ V ⊂ Ad(p ! (A)).
Consider the exact sequence of vector bundles over Σ: This induces a linear splitting (hor, h) : p * Ad(A) → Ad(p ! (A)) to the exact sequence above, and we define metrics g Σ on T Σ and g p ! A on p ! (A) so that be isometries. The metric g = (g p ! (A) , g Σ ) on Ad(p ! (A)) is the one in the output of our recipe. The construction of ∇ which satisfies ( * * ), on the other hand, is subtler, and proceeds in steps.
Step One. First consider the Riemannian connection ∇ R : T Σ T Σ of g Σ , which satisfies Step Two. Let the horizontal-and vertical projections corresponding to g Σ be denoted by P H , P V : T Σ → T Σ, and define a new connection . Note that V, H ⊂ T Σ are subconnections by construction. We claim that ∇ Σ is g Σ -metric, ∇ Σ = ∇ Σ,gΣ . Indeed, note that by definition of ∇ Σ , we have Step Three. There exist unique C ∞ (Σ)-linear maps D : Γ(H) → End(Γ(V )), E : Γ(H) −→ End(Γ(C)), satisfying the Leibniz rule , w ∈ Γ(H) and α ∈ Γ(C), and such that  Step Four. Let now ∇ : T Σ p ! (A) be the connection which satisfies for all v, v ′ ∈ Γ(V ), w ∈ Γ(H) and α ∈ Γ(C), and where c denotes the extension of to a form c ∈ Γ(∧ 2 p * (T * M ) ⊗ V ). This concludes our recipe and all there is left to do is to check that ( * * ) is satisfied.
This shows that ( * * ) holds true, and concludes the proof that char(p ! (A)) = p * char(A).