Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free $\mathcal O_X$-module $\mathcal A$ which has a Lie algebroid structure. Here $X$ is either a complex manifold or a regular scheme over an algebraically closed field $k$. One spectral sequence can be associated with $\mathcal A$ by choosing a global section $V$ of $\mathcal A$, and considering a Koszul complex with a differential given by inner product by $V$. This spectral sequence is shown to degenerate at the second page by using Deligne's degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $\mathcal D_E$ of a holomolorphic vector bundle $E$ on a complex manifold. If $V$ is a differential operator on $E$ with scalar symbol, i.e, a global section of $\mathcal D_E$, we associate with the pair $(E,V)$ a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($E=0$) case


Introduction
In this paper we consider some spectral sequences that one can attach to a Lie algebroid.
To be more precise, if X is a complex manifold, or a regular noetherian scheme over an algebraically closed field k of characteristic zero, we consider a locally free O X -module A having a Lie algebroid structure (definitions will be given in the next Section). One can introduce a complex Ω • A = Λ • A * which is a generalization of the (holomorphic) de Rham complex Ω • X . Now a Lie algebroid A comes with a morphism of sheaves of Lie k-algebras (the anchor morphism) to the tangent sheaf Θ X , and the kernel of the anchor is a sheaf of ideals of A (and a sheaf of Lie O X -algebras); this allows one to introduce, in analogy with the Hochschild-Serre spectral sequence [14], a filtration leading to a spectral sequence which converges to the hypercohomology H(X, Ω • A ). This was already considered in [3] in the C ∞ case; moreover, [17,16] describe this spectral sequence in the case of the Atiyah algebroid of a vector bundle. In [4] and [5] this and other spectral sequences were studied in detail. Lie-Rinehart algebras can be regarded as special cases of Lie algebroids, so that we get a spectral sequence for Lie-Rinehart algebras: this generalizes the Hochschild-Serre spectral sequence for ideals in Lie algebras [14].
Other spectral sequences arise when we fix a section V of A ; this yields a complex of the Koszul type, which we call a Lie-Koszul complex. Then the general machinery of homological algebra [13,18] produces two spectral sequences. In Section 2, by using Deligne's degeneracy criterion [11], we show that the second spectral sequence degenerates. The fact that this spectral sequence satisfies the condition of Deligne's criterion means that the Lie-Koszul complex of a (holomorphic) Lie algebroid is formal (it is isomorphic, in the derived category of coherent sheaves, with the complex formed by its cohomology sheaves).
To study the first spectral sequence of a Lie-Koszul complex, we specialize to the case when A is the Atiyah algebroid of a holomorphic vector bundle E on a complex manifold X (Section 3). Let us recall that D E is the bundle of first order differential operators on E with scalar symbol. D E sits in an exact sequence of sheaves of O X -modules where σ is the symbol map. This spectral sequence relates to the twisted holomorphic equivariant cohomology we introduced in [6]. For E = 0 (i.e, in the case of the de Rham complex) this spectral sequence was studied by Carrell and Lieberman [8] and Bismut [2] when X is Kähler manifold. In that case the spectral sequence degenerates at the first page.

Formality of the Lie-Koszul complexes
We consider a (holomorphic) Lie algebroid A , over X, the latter being a complex manifold, or a regular noetherian scheme over an algebraically closed field k. We choose a global section V of A and consider the morphism (inner product) the Lie-Koszul complex associated with the pair (A , V ). This generalizes the Koszul complex (Ω −• X , i V ) associated with the complex of differential forms on X with the differential given by the inner product by a (holomorphic) vector field V on X. This will be called the de Rham-Koszul complex associated with the vector field V .
By general principles [13,18] we can associate two spectral sequences with this complex, both converging to the hypecohomology H(K • A , i V ). In general, if A, B are Abelian categories, denote by D + (A) the derived category of complexes of objects in A bounded from below, and let F : D + (A) → B a cohomological functor. 1 Let K be an object in D + (A). We recall from [13,18] that with these data one can associate two spectral sequences, both functorial in K , and both converging to R • F (K ). The first two pages of the first spectral sequence are ) and the differential d 1 coincides (perhaps up to a sign, depending on conventions) with the differential of the complex K . The second page of the second spectral sequence is The degeneration of the the second spectral sequence may be studied by means of Deligne's degeneracy criterion [11]. Let us state it in generality. We shall replace the derived category D + (A) by the bounded derived category D b (A). (i) the spectral sequence II • degenerates at its second page for every choice of the functor F ; (In the language of homological algebra, the second condition is called formality of the complex K • .) To apply Deligne's criterion to our case we take A = Coh(X), B = K(Ab) (the category of complexes of Abelian groups) and for F we take the global section functor Γ. The object 1 F is said to be a cohomological functor if it maps every distinguished triangle to a long exact sequence We denote by H • A the cohomology sheaves of the complex (K • A , i V ), and by Y the scheme of zeroes of V . It is a closed, possibly nonreduced, subscheme (analytical subspace) of X. The sheaves H • A are supported on Y . Let j : Y → X be the scheme-theoretic inclusion, or the inclusion as a morphism in the category of analytic spaces (a closed immersion). The functor j * is right adjoint to j * , so that there are morphisms j ⋆ : F → j * j * F for every coherent sheaf F on X. There is a commutative diagram As a consequence, the objects (K • A , i V ) and i H i A [−i] are isomorphic in the derived category D b (X). By Deligne's degeneracy criterion, we obtain that the spectral sequence II • degenerates at the second page.
We can also say something about the hypercohomology H • (K A ). Let us denote by dim Y the dimension of the highest-dimensional component of Y . The proof of the following result goes as in the case of the de Rham-Koszul complex treated in [8], p. 306.
Proof. Where V = 0 the Lie-Koszul complex is exact, so that the supports of the cohomology sheaves H q A are contained in Y ; hence II p,q 2 = 0 for p > dim Y . Moreover, H q A = 0 for q > 0. Thus II p,q 2 = 0 for p + q > dim Y . By standard homological arguments we get the thesis.
If dim Y = 0, 1, this gives an easy proof of the degeneration of the second spectral sequence at the second page, since d 2 : II p,q 2 → II p+2,q−1 2 vanishes in that case. One also has When dim Y = 0, the second page of the spectral sequence is such that II p,q 2 = 0 if p = 0.

A spectral sequence associated with Atiyah algebroids
In this section we study the spectral sequence I • in the special case when the Lie algebroid A is the Atiyah algebroid D E of a holomorphic vector bundle E on a complex manifold X (as in eq. (1)).
We fix once and for all a section V in Γ(D E ). The pair (E , V ) is called an equivariant holomorphic vector bundle. ("Equivariant" refers to the fact that V covers the infinitesimal action of the vector field σ(V ) on X.) We consider the associated Lie-Koszul complex, i.e., the complex (K • E , i V ) where K p E = Λ −p D * E for p ≤ 0, and K p E = 0 for p > 0. This twisted Koszul complex, or, to be more precise, its Dolbeault resolution, is a building block of a "twisted holomorphic equivariant cohomology" that we introduced in [6] and for which we proved a localization formula that generalizes Carrell-Lieberman's [9,7], Feng-Ma's [12] and Baum-Bott's [1] formulas.
The spectral sequence I • relates in this case to the double complex we introduced in [6]. For E = 0 this spectral sequence was studied by Carrell and Lieberman [8] (see also Bismut [2]) and in turn relates to K. Liu's "untwisted" holomorphic equivariant cohomology [15].
We denote by Ω p,q X the sheaf of differential forms of type (p, q) on X, and consider the complex where by∂ E we collectively denote the Cauchy-Riemann operators of the bundles Λ p D * E . We denote by H • V (X, E ) the cohomology of this complex. For E = 0 this reduces to the cohomology introduced by K. Liu [15] (see also Carrell and Lieberman [8] and Bismut [2].) Proof. The double complex Λ −• D * E ⊗ O X Ω 0,• X is an acyclic resolution of the complex K • E , and the total complex of ( . (This resolution is not made by coherent sheaves, but the argument works anyway, just going into the category of sheaves of Abelian groups.) This filtration of the complex (Q (•) E (X),∂ E ,V ) defines a spectral sequence whose zeroth page is The spectral sequence converges to the cohomology H • V (X, E ). The differential d 0 coincides with∂ E , as one easily checks. Therefore, It is now easy to check that this spectral sequence coincides with I • .
Henceforth we assume that the zero locus Y of V is a complex submanifold of X. Therefore it makes sense to consider the complex (3) on Y ; after lettingẼ = E |Y , we denote this new complex Q •Ẽ (Y ). Denoting by j :Ỹ → X the embedding, we have the restriction morphism j * : , which is a morphism of filtered complexes. We are going to show that, under some conditions, this is a quasi-isomorphism.
Note that there is an exact sequence This operator vanishes on DẼ , so it is well defined on N Y /X . If it is injective, by composing with the projection D E |Y → N Y /X it yields an isomorphism, thus splitting the sequence (4).
For clarity, we stress what we are assuming: The zero locus Y of V is a complex submanifold on X, and the morphism This implies the following preliminary result. LetK • E be the complex of sheaves on Ỹ K p E = Λ −p D * E with the zero differential.
Proof. There is a naturally defined morphism j * H p E →K p E . We need to show that this gives an isomorphism between the stalks of the two sheaves. Considering the exact sequence (1) restricted to the stalks at a point y ∈ Y , it splits, and one has LetṼ be the vector fieldṼ = σ(V ). It vanishes on Y . Then one knows that the cohomology of the complex (Ω −• X , iṼ ) restricted to Y is isomorphic to the cohomology of the complex (Ω −• Y , 0) [2]. This, together with the Künneth theorem, implies the result.
The following result generalizes to the twisted case Theorem 5.1 in [2]. The proof goes as in [2], but for clarity we report it here, adapted to the present situation, and with some more details.
with differentialδ = δ +i V , where δ is the usualČech differential. We define the descending filtration Let (E • (X), d • ) be the ensuing spectral sequence. The d 0 differential acting on the 0-th page coincides with i V , so that the first page of the spectral sequence is The differential d 1 acting on this complex is theČech differential. By Lemma 3.4, we also have whereŨ is the open cover of Y obtained by restricting the open sets of U to Y .
Consider now the complex The resulting spectral sequence E • (Y ) has a vanishing d 0 differential, hence E 1 (Y ) coincides with the E 0 page. The restriction morphism j * induces a morphism j * : E 1 (X) → E 1 (Y ). By the commutativity of the diagram (2), this is an isomorphism and commutes with the respective differentials (which are theČech differentials of the respectiveČech complexes). By [10, Ch. XV, Thm. 3.2] the successive pages of the two spectral sequences are isomorphic, and the spectral sequences converge to the same group. Therefore, the complexes C (•) (X) and C (•) (Y ) are quasi-isomorphic.
Via the standardČech-Dolbeault spectral sequence, the cohomology of the complex C (•) (Y ) is, after taking a direct limit on the covers U, isomorphic to the cohomology of (Q •Ẽ (Y ),∂Ẽ ). In the same way, the cohomology of C (•) (X) is isomorphic, after taking a direct limit, to the cohomology of (Q • E (X),∂ E ,V ). This concludes the proof. Proof. Since V = 0 on Y this follows from Remark 3.1.
Let us eventually consider the first spectral sequence I • . Its first page is I p,q 1 = H q (X, Λ −p D * E ). In the untwisted (E = 0) case, and assuming that X is compact and Kähler, Carrell and Lieberman [8], by an argument inspired by Deligne's degeneracy criterion, show that d 1 = 0, so that this spectral sequence degenerates at the first page.