1 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 \({{\mathscr {O}}}_X\)-module \({{\mathscr {A}}}\) having a Lie algebroid structure (definitions will be given in the next Section). One can introduce a complex \(\Omega _{{\mathscr {A}}}^\bullet =\Lambda ^\bullet {{\mathscr {A}}}^*\) which is a generalization of the (holomorphic) de Rham complex \(\Omega _X^\bullet\). Now a Lie algebroid \({{\mathscr {A}}}\) comes with a morphism of sheaves of Lie k-algebras (the anchor morphism) to the tangent sheaf \(\Theta _X\), and the kernel of the anchor is a sheaf of ideals of \({{\mathscr {A}}}\) (and a sheaf of Lie \({{\mathscr {O}}}_X\)-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 \({\mathbb {H}}(X,\Omega _{{\mathscr {A}}}^\bullet )\). This was already considered in [3] in the \(C^\infty\) case; moreover, [16, 17] 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 \({{\mathscr {A}}}\); 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 Sect. 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 \({{\mathscr {A}}}\) is the Atiyah algebroid of a holomorphic vector bundle \({{\mathscr {E}}}\) on a complex manifold X (Sect. 3). Let us recall that \({{{\mathscr {D}}}_{{{\mathscr {E}}}}}\) is the bundle of first order differential operators on \({{\mathscr {E}}}\) with scalar symbol. \({{{\mathscr {D}}}_{{{\mathscr {E}}}}}\) sits in an exact sequence of sheaves of \({{\mathscr {O}}}_X\)-modules

$$\begin{aligned} 0 \rightarrow {\text{ End }}\,({{\mathscr {E}}}) \rightarrow {{{\mathscr {D}}}_{{{\mathscr {E}}}}}\xrightarrow {\sigma } \Theta _X \rightarrow 0 \end{aligned}$$
(1)

where \(\sigma\) is the symbol map. This spectral sequence relates to the twisted holomorphic equivariant cohomology we introduced in [6]. For \({{\mathscr {E}}}=0\) (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.

2 Formality of the Lie-Koszul complexes

We consider a (holomorphic) Lie algebroid \({{\mathscr {A}}}\), 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 \({{\mathscr {A}}}\) and consider the morphism (inner product) \(i_V:{\mathscr {K}}_{{\mathscr {A}}}^\bullet \rightarrow {\mathscr {K}}_{{\mathscr {A}}}^{\bullet +1}\), where \({\mathscr {K}}_{{\mathscr {A}}}^p = \Omega _{{\mathscr {A}}}^{-p}\), \(p\le 0\). We shall call \(({\mathscr {K}}_{{\mathscr {A}}}^\bullet ,i_V)\) the Lie-Koszul complex associated with the pair \(({{\mathscr {A}}},V)\). This generalizes the Koszul complex \((\Omega _X^{-\bullet },i_V)\) 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 \({\mathbb {H}}({\mathscr {K}}_{{\mathscr {A}}}^\bullet ,i_V)\). In general, if \({\mathfrak {A}}\), \({\mathfrak {B}}\) are Abelian categories, denote by \(D^+({\mathfrak {A}})\) the derived category of complexes of objects in \({\mathfrak {A}}\) bounded from below, and let \(F:D^+({\mathfrak {A}}) \rightarrow {\mathfrak {B}}\) a cohomological functor.Footnote 1 Let \({\mathscr {K}}\) be an object in \(D^+({\mathfrak {A}})\). We recall from [13, 18] that with these data one can associate two spectral sequences, both functorial in \({\mathscr {K}}\), and both converging to \(R^{\bullet }F({\mathscr {K}})\). The first two pages of the first spectral sequence are

$$\begin{aligned} I_1^{p,q}=R^qF({\mathscr {K}}^p),\qquad I_2^{p,q} = H^p(R^qF({\mathscr {K}})) \end{aligned}$$

and the differential \(d_1\) coincides (perhaps up to a sign, depending on conventions) with the differential of the complex \({\mathscr {K}}\). The second page of the second spectral sequence is

$$\begin{aligned} I\! I_2^{p,q} = R^pF(H^q({\mathscr {K}})). \end{aligned}$$

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 \(D^+({\mathfrak {A}})\) by the bounded derived category \(D^b({\mathfrak {A}})\).

Theorem 2.1

(Deligne) The following two conditions are equivalent:

  1. (i)

    the spectral sequence \(I\! I_\bullet\) degenerates at its second page for every choice of the functor F;

  2. (ii)

    \({\mathscr {K}}^\bullet\) is isomorphic to \(\oplus _i H^i({\mathscr {K}}^\bullet )[-i]\) in \(D^b({\mathfrak {A}})\).

(In the language of homological algebra, the second condition is called formality of the complex \({\mathscr {K}}^\bullet\).)

To apply Deligne’s criterion to our case we take \({\mathscr {A}} = Coh(X)\), \({\mathscr {B}} = {{\mathbf {K}}}({\text{ Ab}})\) (the category of complexes of Abelian groups) and for F we take the global section functor \(\Gamma\). The object we fix in \(D^b(X)\) is the Lie-Koszul complex \(({\mathscr {K}}_{{\mathscr {A}}}^\bullet ,i_{ V})\). We denote by \({\mathscr {H}}_{{\mathscr {A}}}^\bullet\) the cohomology sheaves of the complex \(({\mathscr {K}}_{{\mathscr {A}}}^\bullet ,i_{ V})\), and by Y the scheme of zeroes of V. It is a closed, possibly nonreduced, subscheme (analytical subspace) of X. The sheaves \({\mathscr {H}}_{{\mathscr {A}}}^\bullet\) are supported on Y. Let \(j :Y \rightarrow X\) 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 \(j^\star :{\mathscr {F}}\rightarrow j_*j^*{\mathscr {F}}\) for every coherent sheaf \({\mathscr {F}}\) on X. There is a commutative diagram

(2)

i.e., \(j^\star\) is a morphism of complexes if we equip \(j_*j^*{\mathscr {K}}_{{\mathscr {A}}}^\bullet\) with the zero morphisms. Finally, \({\mathscr {H}}_{{\mathscr {A}}}^\bullet \simeq j_*j^*{\mathscr {K}}_{{\mathscr {A}}}^\bullet\). Now we have:

Proposition 2.2

The morphism of complexes \(j^\star :({\mathscr {K}}_{{\mathscr {A}}}^{\bullet }, i_{V}) \rightarrow (j_*j^*{\mathscr {K}}_{{\mathscr {A}}}^{\bullet },0)\) is a quasi-isomorphism.

As a consequence, the objects \(({\mathscr {K}}_{{\mathscr {A}}}^{\bullet }, i_{V})\) and \(\bigoplus _i {\mathscr {H}}_{{\mathscr {A}}}^i[-i]\) are isomorphic in the derived category \(D^b(X)\). By Deligne’s degeneracy criterion, we obtain that the spectral sequence \(I\! I_\bullet\) degenerates at the second page.

We can also say something about the hypercohomology \({\mathbb {H}}^\bullet ({\mathscr {K}}_{{\mathscr {A}}})\). 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.

Proposition 2.3

\({\mathbb {H}}^m({\mathscr {K}}_{{\mathscr {A}}}^\bullet ,i_V)=0\) for \(m>\dim Y\).

Proof

Where \(V\ne 0\) the Lie-Koszul complex is exact, so that the supports of the cohomology sheaves \({\mathscr {H}}_{{\mathscr {A}}}^q\) are contained in Y; hence \(I\! I_2^{p,q}=0\) for \(p>\dim Y\). Moreover, \({\mathscr {H}}_{{\mathscr {A}}}^q=0\) for \(q>0\). Thus \(I\! I_2^{p,q}=0\) for \(p+q>\dim Y\). By standard homological arguments we get the thesis. \(\square\)

If \(\dim Y=0,1\), this gives an easy proof of the degeneration of the second spectral sequence at the second page, since \(d_2:I\! I_2^{p,q} \rightarrow I\! I_2^{p+2,q-1}\) vanishes in that case. One also has

$$\begin{aligned} {\mathbb {H}}^m({\mathscr {K}}^\bullet _{{\mathscr {A}}},i_V) \simeq \bigoplus _{p+q=m} H^p(X,{\mathscr {H}}^q_{{\mathscr {A}}}). \end{aligned}$$

When \(\dim Y=0\), the second page of the spectral sequence is such that \(I\! I_2^{p,q} =0\) if \(p\ne 0\).

3 A spectral sequence associated with Atiyah algebroids

In this section we study the spectral sequence \(I_\bullet\) in the special case when the Lie algebroid \({{\mathscr {A}}}\) is the Atiyah algebroid \({{{\mathscr {D}}}_{{{\mathscr {E}}}}}\) of a holomorphic vector bundle \({{\mathscr {E}}}\) on a complex manifold X (as in eq. (1)).

We fix once and for all a section V in \(\Gamma ({{{\mathscr {D}}}_{{{\mathscr {E}}}}})\). The pair \(({{\mathscr {E}}},V)\) is called an equivariant holomorphic vector bundle. (“Equivariant” refers to the fact that V covers the infinitesimal action of the vector field \(\sigma (V)\) on X.) We consider the associated Lie-Koszul complex, i.e., the complex \(({\mathscr {K}}_{{\mathscr {E}}}^\bullet ,i_{V})\) where \({\mathscr {K}}_{{\mathscr {E}}}^{p} = \Lambda ^{-p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\) for \(p\le 0\), and \({\mathscr {K}}_{{\mathscr {E}}}^{p}=0\) 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 [7, 9], Feng-Ma’s [12] and Baum-Bott’s [1] formulas.

The spectral sequence \(I_\bullet\) relates in this case to the double complex we introduced in [6]. For \({{\mathscr {E}}}=0\) 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 \(\Omega _X^{p,q}\) the sheaf of differential forms of type (p, q) on X, and consider the complex

$$\begin{aligned} Q^k_{{{\mathscr {E}}}} (X)= \bigoplus _{q-p=k}\Gamma \left[ \Lambda ^{p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,q}_X\right] \end{aligned}$$
(3)

with the differential \({\bar{\partial }}_{{{\mathscr {E}}},V}={\bar{\partial }}_{{\mathscr {E}}}+ i_{V}\), where by \({\bar{\partial }}_{{\mathscr {E}}}\) we collectively denote the Cauchy-Riemann operators of the bundles \(\Lambda ^{p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\). We denote by \(H^\bullet _{V}(X,{{\mathscr {E}}})\) the cohomology of this complex. For \({{\mathscr {E}}}=0\) this reduces to the cohomology introduced by K. Liu [15] (see also Carrell and Lieberman [8] and Bismut [2].)

Remark 3.1

If \(V=0\) then \(H^k_{V}(X,{{\mathscr {E}}})=\bigoplus _{q-p=k} H^q(X,{\Lambda ^{p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}})\).

Proposition 3.2

The cohomology \(H^\bullet _{V}(X,{{\mathscr {E}}})\) is isomorphic to the hypercohomology \({\mathbb {H}}^\bullet ({\mathscr {K}}_{{\mathscr {E}}}^\bullet )\) of the complex \(({\mathscr {K}}_{{\mathscr {E}}}^\bullet ,i_{V})\).

Proof

The double complex \(\Lambda ^{-\bullet }{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,\bullet }_X\) is an acyclic resolution of the complex \({\mathscr {K}}_{{\mathscr {E}}}^\bullet\), and the total complex of \((\Lambda ^{-\bullet }{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,\bullet }_X,i_V,{\bar{\partial }}_{{\mathscr {E}}})\) coincides with \(({\Lambda ^{\bullet }{{{\mathscr {D}}}_{{\mathscr {E}}}^*}},{\bar{\partial }}_{{{\mathscr {E}}},V})\). (This resolution is not made by coherent sheaves, but the argument works anyway, just going into the category of sheaves of Abelian groups.) \(\square\)

We denote

$$\begin{aligned} G_{p}^q= \bigoplus _{0\le p'\le -p} \Gamma \left[ \Lambda ^{p'}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,q}_X\right] \end{aligned}$$

with \(p\le 0\), so that \(G_\bullet ^q\) is a descending filtration of \(Q^q_{{\mathscr {E}}}(X)\). Note that

$$\begin{aligned} G_p^q = G_p\cap Q^q_{{\mathscr {E}}}(X)\qquad {\text{ and} }\qquad G_p^{p+q}/G_{p+1}^{p+q} = \Gamma \left[ \Lambda ^{-p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,q}_X\right] . \end{aligned}$$

This filtration of the complex \((Q_{{\mathscr {E}}}^{(\bullet )}(X) ,{\bar{\partial }}_{{{\mathscr {E}}},V})\) defines a spectral sequence whose zeroth page is

$$\begin{aligned} E_0^{p,q} = G_p^{p+q}/G_{p+1}^{p+q} = \Gamma \left[ \Lambda ^{-p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,q}_X\right] . \end{aligned}$$

The spectral sequence converges to the cohomology \(H^\bullet _{V}(X,{{\mathscr {E}}})\). The differential \(d_0\) coincides with \({\bar{\partial }}_{{\mathscr {E}}}\), as one easily checks. Therefore,

$$\begin{aligned} E_1^{p,q} = H^q(E_0^{p,\bullet }, d_0) = H^q(\Gamma [\Lambda ^{-p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}\otimes _{{{\mathscr {O}}}_X}\Omega ^{0,\bullet }_X],{\bar{\partial }}_{{\mathscr {E}}}) \simeq H^q(X,{\Lambda ^{-p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}}). \end{aligned}$$

It is now easy to check that this spectral sequence coincides with \(I_\bullet\).

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 letting \({\tilde{{{\mathscr {E}}}}}={{\mathscr {E}}}_{\vert Y}\), we denote this new complex \(Q^\bullet _{{\tilde{{{\mathscr {E}}}}}}(Y)\). Denoting by \(j:{{\tilde{Y}}}\rightarrow X\) the embedding, we have the restriction morphism \(j^*:Q^ \bullet _{{{\mathscr {E}}}} (X)\rightarrow Q^\bullet _{{\tilde{{{\mathscr {E}}}}}}( Y)\), 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

$$\begin{aligned} 0 \rightarrow {\mathscr {D}}_{{\tilde{{{\mathscr {E}}}}}} \rightarrow {{{\mathscr {D}}}_{{{\mathscr {E}}}}}_{\vert Y} \rightarrow N_{Y/X}\rightarrow 0 \end{aligned}$$
(4)

where \(N_{Y/X}\) is the normal bundle to Y. Since V is zero on Y, the commutator \({\mathbb {L}}_{V}(u) = [V,u]\) is well defined if \(u\in {{{\mathscr {D}}}_{{{\mathscr {E}}}}}_{\vert Y}\). This operator vanishes on \({\mathscr {D}}_{{\tilde{{{\mathscr {E}}}}}}\), so it is well defined on \(N_{Y/X}\). If it is injective, by composing with the projection \({{{\mathscr {D}}}_{{{\mathscr {E}}}}}_{\vert Y} \rightarrow N_{Y/X}\) it yields an isomorphism, thus splitting the sequence (4).

For clarity, we stress what we are assuming:

Assumption 3.3

The zero locus Y of V is a complex submanifold on X, and the morphism \({\mathbb {L}}_{V}:N_{Y/X} \rightarrow {{{\mathscr {D}}}_{{{\mathscr {E}}}}}_{\vert Y}\) is injective.

This implies the following preliminary result. Let \(\tilde{{\mathscr {K}}}^\bullet _{\tilde{{{\mathscr {E}}}}}\) be the complex of sheaves on Y

$$\begin{aligned} \tilde{{\mathscr {K}}}^p_{\tilde{{{\mathscr {E}}}}}=\Lambda ^{-p}{\mathscr {D}}_{\tilde{{{\mathscr {E}}}}}^*\end{aligned}$$

with the zero differential.

Lemma 3.4

\(j^*{\mathscr {H}}^p_{{\mathscr {E}}}\simeq \tilde{{\mathscr {K}}}^p_{\tilde{{{\mathscr {E}}}}}\). In particular, \({\mathscr {H}}^p_{{\mathscr {E}}}=0\) if \(-p>\dim Y\).

Proof

There is a naturally defined morphism \(j^*{\mathscr {H}}^p_{{\mathscr {E}}}\rightarrow \tilde{{\mathscr {K}}}^p_{\tilde{{{\mathscr {E}}}}}\). 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 \(y\in Y\), it splits, and one has

$$\begin{aligned}({{\mathscr {K}}}^p_{{{{\mathscr {E}}}}})_y \simeq \bigoplus _{q+q'=-p} (\Omega ^q_X)_y \otimes \Lambda ^{q'} ({\text{ End }}\,({{\mathscr {E}}}))_y\,\\ (\tilde{{\mathscr {K}}}^p_{{{\tilde{{{\mathscr {E}}}}}}})_y \simeq \bigoplus _{q+q'=-p} (\Omega ^q_Y)_y \otimes \Lambda ^{q'} ({\text{ End }}\,({{\mathscr {E}}}))_y\,,\end{aligned}$$

Let \({{\tilde{V}}}\) be the vector field \({{\tilde{V}}}=\sigma (V)\). It vanishes on Y. Then one knows that the cohomology of the complex \((\Omega ^{-\bullet }_X,i_{{{\tilde{V}}}})\) restricted to Y is isomorphic to the cohomology of the complex \((\Omega ^{-\bullet }_Y,0)\) [2]. This, together with the Künneth theorem, implies the result. \(\square\)

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.

Theorem 3.5

Under the Assumption 3.3,the restriction morphism \(j^*:Q^ \bullet _{{{\mathscr {E}}}} (X)\rightarrow Q^\bullet _{{\tilde{{{\mathscr {E}}}}}}(Y)\) is a quasi-isomorphism.

Proof

Let \({\mathfrak {U}}\) be an open cover of X, and consider the Čech-Koszul complex

$$\begin{aligned} C^{(k)}(X) = \bigoplus _{p+q=k}{\check{C}}^p({\mathfrak {U}},{\mathscr {K}}_{{\mathscr {E}}}^q) \end{aligned}$$

with differential \({\tilde{\delta }}=\delta +i_V\), where \(\delta\) is the usual Čech differential. We define the descending filtration

$$\begin{aligned} F_q = \bigoplus _{\begin{array}{c} p'\ge p \\ q \end{array}} {\check{C}}^{p'}({\mathfrak {U}},{\mathscr {K}}_{{\mathscr {E}}}^q),\qquad F_p^q = F_p\cap C^{(q)}(X) \end{aligned}$$

so that \(F_{q+1}^p\subset F_q^p\),

$$\begin{aligned} F_{q}^{p+q}/F_{q+1}^{p+q}={\check{C}}^p({\mathfrak {U}},{\mathscr {K}}_{{\mathscr {E}}}^q), \end{aligned}$$

and

$$\begin{aligned} {\tilde{\delta }}(F_q^p) \subset F_{q+1}^{p+1} + F_{q}^{p+1} = F_{q}^{p+1}. \end{aligned}$$

Let \((E_\bullet (X),d_\bullet )\) 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

$$\begin{aligned} E_1(X)^{p,q}= {\check{C}}^p({\mathfrak {U}},{\mathscr {H}}_{{\mathscr {E}}}^q). \end{aligned}$$

The differential \(d_1\) acting on this complex is the Čech differential. By Lemma 3.4, we also have

$$\begin{aligned} E_1(X)^{p,q}\simeq {\check{C}}^p(\tilde{{\mathfrak {U}}},\tilde{{\mathscr {K}}}_{{\tilde{{{\mathscr {E}}}}}}^q) \end{aligned}$$

where \(\tilde{{\mathfrak {U}}}\) is the open cover of Y obtained by restricting the open sets of \({\mathfrak {U}}\) to Y.

Consider now the complex

$$\begin{aligned} C^{(k)}(Y) = \bigoplus _{p+q=k}{\check{C}}^p(\tilde{{\mathfrak {U}}},\tilde{{\mathscr {K}}}^q_{\tilde{{{\mathscr {E}}}}}). \end{aligned}$$

The resulting spectral sequence \(E_\bullet (Y)\) has a vanishing \(d_0\) differential, hence \(E_1(Y)\) coincides with the \(E_0\) page. The restriction morphism \(j^*\) induces a morphism \(j^*:E_1(X) \rightarrow E_1(Y)\). By the commutativity of the diagram (2), this is an isomorphism and commutes with the respective differentials (which are the Čech differentials of the respective Čech 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 \(C^{(\bullet )}(X)\) and \(C^{(\bullet )}(Y)\) are quasi-isomorphic.

Via the standard Čech-Dolbeault spectral sequence, the cohomology of the complex \(C^{(\bullet )}(Y)\) is, after taking a direct limit on the covers \({\mathfrak {U}}\), isomorphic to the cohomology of \((Q^{\bullet }_{\tilde{{{\mathscr {E}}}}}(Y),{\bar{\partial }}_{{\tilde{{{\mathscr {E}}}}}})\). In the same way, the cohomology of \(C^{(\bullet )}(X)\) is isomorphic, after taking a direct limit, to the cohomology of \((Q_{{\mathscr {E}}}^{\bullet }(X),{\bar{\partial }}_{{{\mathscr {E}}},V})\). This concludes the proof.\(\square\)

Corollary 3.6

\(H^k_{V}(X,{{\mathscr {E}}})\simeq \bigoplus _{q-p=k}H^q( Y,\Lambda ^p{\mathscr {D}}_{{\tilde{{{\mathscr {E}}}}}}^*)\).

(Compare with Remark 3.1.)

Proof

Since \(V=0\) on Y this follows from Remark 3.1.\(\square\)

Let us eventually consider the first spectral sequence \(I_\bullet\). Its first page is

$$\begin{aligned} I_1^{p,q}=H^q(X,\Lambda ^{-p}{{{\mathscr {D}}}_{{\mathscr {E}}}^*}). \end{aligned}$$

In the untwisted (\({{\mathscr {E}}}=0\)) 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.