FRÖLICHER SPECTRAL SEQUENCE AND HODGE STRUCTURES ON THE COHOMOLOGY OF COMPLEX PARALLELISABLE MANIFOLDS

For complex parallelisable manifolds Γ\G, with G a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.


Introduction
This article is mainly concerned with the cohomology of complex manifolds of the form X = Γ\G, where G is a connected complex Lie group and Γ ⊆ G a cocompact, discrete subgroup.Every such manifold has trivial holomorphic tangent bundle and vice versa Wang [19], cf. also [20,Section 1.12], has shown that every compact complex parallelisable manifold is of this form and we may assume G to be simply connected.
We consider the Frölicher spectral sequence of a compact complex parallelisable manifold X = Γ\G.If the Frölicher spectral sequence of a compact complex manifold degenerates at the first page, then every holomorphic form is closed.By this fact, we can easily say that the Frölicher spectral sequence of X degenerates at the first page if and only if G is abelian.Hence, unlike compact Kähler manifolds, if G is not abelian X does not satisfy ∂ ∂-Lemma as in [6].
If G is nilpotent, the Dolbeault cohomology and de Rham cohomology can be computed from the Lie algebra of G [16].In particular, it is independent of the lattice Γ.Furthermore, it was recently shown in [13] by Popovici, Ugarte and the second named author that they are page-1-∂ ∂-manifolds, meaning that the Frölicher spectral sequence degenerates at the second page and the Hodge filtration induces a pure Hodge structure on the de Rham cohomology.One of the goals of this article is to understand to what extend these results hold for not necessarily nilpotent G.
Unlike in the nilpotent case, in general the dimensions of Dolbeault and de Rham cohomology can depend on the lattice (see e.g.[3,Ex. 3.4] for G solvable and [20] or [7] for G = SL 2 (C)).Nevertheless, it was shown by the first-named author in [10] that, for G a solvable complex Lie group, regardless of the choice of Γ, the Frölicher spectral sequence of degenerates at the second page.Our first result generalizes this by also asserting the existence of a pure Hodge structure on the cohomology of these manifolds.
The proof will be given in section 3. Similar techniques also yield an analogue of this theorem for certain solvmanifolds of splitting type.(see Thm. 9).
In the semisimple case, the situation is more subtle.Computations by Ghys [7] and Winkelmann [20] show that for G = SL 2 (C), there exist lattices Γ, Γ ′ ⊆ G s.t.Γ\G is page-1-∂ ∂ and Γ ′ \G is not.Building on work of Akhiezer, we give a conceptual explanation of this phenomenon.
Suppose G is semi-simple.Take a maximal compact subgroup K ⊂ G. Denote by g and k the Lie algebra of G and K respectively.Then, associated with the locally symmetric space Y = Γ\G/K, we have the canonical injection H k (g; k, C) ֒→ H k (Γ, C) where H k (g, k, C) is the relative Lie algebra cohomology and H k (Γ, C) is the group cohomology (see [17,Introduction] for instance).In general, this injection is not an isomorphism.
Theorem B. Let G be semisimple.
1.The Frölicher spectral sequence of X degenerates at the second page.

X is a page-1-∂ ∂ manifold if and only if H
In view of Theorems A and B, it appears natural to make the following: Conjecture.For any compact complex parallelisable manifold X, the Frölicher spectral sequence degenerates at the second page.
Apart from the results in this article, indication that this might hold is given by [11], where it is shown that d 2 : E 0,1 2 (X) → E 2,0 2 (X) always vanishes.

Page-1-∂ ∂-Manifolds
We briefly recall some of the terminology of [13].Let A = (A• , •,∂, ∂) be a bounded double complex of complex vector spaces and TotA• := p+q= • A p,q the associated total complex with differential d := ∂ + ∂.Associated with the filtration F r A = p≥r A p,q we get an induced filtration on the total cohomology H• dR (A) := H•(TotA), still denoted by F , and the spectral sequence Analogously, for the filtration F r A = q≥r A p,q we obtain a second spectral sequence converging to H dR (A), which we denote by Ēr (A).It has the spaces H p,q ∂ (A) on its first page.The Bott-Chern cohomology is defined as and the Aeppli cohomology as All double complexes which we will consider will have finite dimensional cohomology, i.e. that the spaces H p,q ∂ (A) and H p,q ∂ (A) are finite dimensional for all p, q ∈ Z (this also implies that E p,q r (A), Ēp,q r (A), H p,q BC (A), H p,q A (A), H k dR (A) are finite dimensional [18, sect.2]).Moreover, as the notation already suggests, in most cases we will consider double complexes A that are equipped with a real structure, i.e. an antilinear involution (.) s.t.A p,q = A q,p and ∂ā = ∂a.In this case the second spectral sequence is determined by the first and can be ignored.We denote by H p,q (A) := im(H p,q BC (A) −→ H p+q dR (A)).We have equalities Recall [6] that a double complex A as above is said to satisfy the ∂ ∂-Lemma (or to have the ∂ ∂-property) if it satisfies one (hence all) of the following equivalent properties: Both spectral sequences degenerate at page 1 and for all k ∈ Z, the filtrations F and F on H k dR (A) induce a pure Hodge structure of degree k, i.e.H k dR (A) = p+q=k H p,q (A).
3. Every class in H p,q ∂ (A) admits a d-closed pure-type representative and the map H p,q ∂ (A) −→ H p,q (A) sending a class via such a representative induces a well-defined isomorphism p,q∈Z (and analogously for H p,q ∂ (A)).4. A is a direct sum of complexes of the following types (a) 'squares': complexes with a single pure-bidegree generator a, s.t.
∂ ∂a = 0 ∂a ∂ ∂a a ∂a (b) 'dots': complexes concentrated in a single bidegree, with all differentials being zero.a The main interest of this paper is following analogue of this property, introduced in [13].
Definition 1.A bounded double complex A is said to have the page-1-∂ ∂property if both its spectral sequences degenerate at page 2 and for all k ∈ Z, the filtrations F and F on H k dR (A) induce a pure Hodge structure of degree k.There is an obvious extension of this notion to any page of the Frölicher spectral sequence, including the usual ∂ ∂-property as the page-0-∂ ∂-property, but we will not need this here. Define

Proposition 2 ([13]
).Let A be a bounded double complex with finite-dimensional cohomology.The following assertions are equivalent:

Every class in E p,q
2 (A) admits a d-closed pure-type representative and the map E p,q 2 (A) → H p,q (A) sending a class via such a representative induces a well-defined linear isomorphism and analogously for Ē2 .

for every r ∈ Z the equality h r
A is a direct sum of squares, dots and 'lines', i.e. complexes generated by a single pure-bidegree generator a, s.t.exactly one of ∂a or ∂a is nonzero.
There is also a characterisation in terms of suitable exactness properties (c.f.[14] for details).
Remark 3. Note that for complexes with real structure property 2 implies the symmetry dim E p,q 2 = dim E q,p 2 .
Let X be a complex manifold and A X = (A • , • X , ∂, ∂) the Dolbeault double complex, i.e. the double complex of C-valued differential forms.We denote E p,q r (X) = E p,q r (A X ) and call it the Frölicher spectral sequence.In this case the total cohomology is the de Rham cohomology Proposition 2 and [18, Cor.13] (c.f. also [2], [3], which treat all cases relevant for us) imply the following.
Corollary 5. Let X be a compact complex manifold.Given any double complex with C and a map C → A X of double complexes, such that for all p, q ∈ Z the induced map If C is equipped with a real structure and ϕ respects this map, it is sufficient to consider only the induced map in H∂ .
The following observation gives rise to many examples of page-1-∂ ∂-complexes.
The spaces E p,q 2 (A⊗B) = H p,q (A⊗B) are identified with H p (A)⊗H q (B).

Complex Lie Groups and Lattices
Let G be a complex Lie group and g its Lie algebra.The complex structure of G induces a splitting g C = g 1,0 ⊕ g 0,1 into i and −i Eigenspaces and we denote by g ∨ C = g 1,0 ⊕ g 0,1 the splitting of the dual space.Denote by Λ g := Λ•g ∨ C ⊆ A G the space of left invariant forms on G.The restriction of the exterior differential makes it into a sub-double complex of A G .Denote by Λ + g = (Λ•g 1,0 , ∂) and Λ − g = (Λ•g 0,1 , ∂) the simple complexes of left-invariant forms of types ( • , 0) and (0, • ).The following observation was made in [13], (formulated there for G nilpotent): Lemma 7. The complex Λ g is the tensor product of the simple complexes Λ + g and Λ − g , hence it is a page-1-∂ ∂-complex.Now assume that there exists a lattice Γ ⊆ G (which, for the purpose of this article, will mean that Γ is a discrete, cocompact subgroup).We will write X := Γ\G for the compact complex manifold obtained as the quotient of G by the left action of Γ.Since Λ g consists of left-invariant forms, it can also be considered as a sub-double complex of A X .It carries the induced real structure.Hence, by Lemma 7 and Corollary 5 we have: In [16], Sakane proved that if G is nilpotent, then the inclusion i : If G is not nilpotent the inclusion i : Λ g → A X does not induce an isomorphism H p,q ∂ (Λ g ) ∼ = H p,q ∂ (X) (e.g.Nakamura manifolds).Since it is known that Lie-groups admitting a lattice are unimodular [12,Lem. 6.2], by averaging associated with a bi-invariant Haar measure, we obtain a map of double complexes µ : A * X → Λ g such that µ • i = Id (see the proof of [4,Theorem 7]).Thus, one obtains a direct sum decomposition A X = Λ g ⊕ A X /Λ g and a corresponding decomposition on all cohomologies considered.In particular, the cohomology groups of Λ g inject into those of X.

Solvable Case
We keep the notation of the preceding section and assume that G is a complex solvable group.Let N be the nilradical of G.We can take a connected simplyconnected complex nilpotent subgroup C ⊆ G such that G = C • N .Associated with C, we define the diagonalizable representation where (Ad c ) s is the semi-simple part of the Jordan decomposition of the adjoint operator (Ad c ) ∈ Aut(g 1,0 ).Denote this representation by Ad s : G → Aut(g 1,0 ).

Then the inclusion B•
).Consider the following subcomplex of A X :
Since we have α −1 dα ∈ V 1 for any holomorphic character α, we have d = ∂ : αV λ → αV λ and also d = ∂ : ᾱVλ → ᾱVλ for any α, λ.Thus, each factor of the direct sum of C• 1 , • 2 is the tensor product of simple complexes, hence it is a page-1-∂ ∂-complex by Lemma 6.This implies Theorem A.

Variant for Complex Solvmanifolds of Splitting Type
In this subsection only, G will not be a complex Lie group, but we assume that G is the semi-direct product C n ⋉ φ N so that: 1. N is a connected simply-connected 2m-dimensional nilpotent Lie group endowed with an N -bi-invariant complex structure J N ; (denote the Lie algebras of C n and N by a and, respectively, n;) 2. for any t ∈ C n , it holds that φ(t) ∈ Aut(N ) is a holomorphic automorphism of N with respect to J N ; 3. φ induces a semi-simple action on n; 4. G has a lattice Γ; (then Γ can be written as Γ = Γ C n ⋉ φ Γ N such that Γ C n and Γ N are lattices of C n and, respectively, N , and, for any Consider the solvmanifold X = Γ\G.This satisfies [8, Assumption 1.1] ([3, Assumption 2.11]).
This result is a generalization of the observation on completely solvable Nakamura manifolds in [13,Corollary 3.3].
Proof.Consider the standard basis {X 1 , . . ., X n } of C n .Consider the decomposition n ⊗ R C = n 1,0 ⊕ n 0,1 induced by J N .By the condition 2, this decomposition is a direct sum of C n -modules.By the condition 3, we have a basis {Y 1 , . . ., Y m } of n 1,0 and characters α 1 , . . ., α m ∈ Hom(C n ; C * ) such that the induced action φ on n 1,0 is represented by For any j ∈ {1, . . ., m}, since Y j is an N -left-invariant (1, 0)-vector field on N , the (1, 0)-vector field α j Y j on C n ⋉ φ N is G-left-invariant.Consider the Lie algebra g of G and the decomposition g C := g ⊗ R C = g 1,0 ⊕ g 0,1 induced by J. Hence we have a basis {X 1 , . . ., X n , α 1 Y 1 , . . ., α m Y m } of g 1,0 , and let x 1 , . . ., x n , α −1 1 y 1 , . . ., α −1 m y m be its dual basis of g 1,0 .Then we have for the C n -action.Then we have where β ∈ Hom(C n ; C * ) and γ ∈ Hom(C n ; C * ) are unitary characters on C n such that αβ −1 and λγ −1 are holomorphic.We have Thus each each factor of the direct sum of C• , • is the tensor product of simple complexes, hence it is a page-1-∂ ∂-complex.

Semisimple Case
In this section, we keep the notation from section 2 and assume G to be a complex semisimple Lie group, and K ⊆ G a maximal compact subgroup.Denote by g = Lie(G), k = Lie(K) the respective Lie algebras.Note that we have g = k⊕Jk.
As before, we denote by X := Γ\G the quotient by the left action of Γ.Later, we will also consider Y := G/K the quotient of the right action by K and Z := Γ\G/K the quotient by both.A first observation is: Proof.This is a consequence of the identification of complex Lie algebras g 1,0 ) and the fact that the de Rham cohomology of compact Lie groups can be computed from left-invariant forms.
Theorem 11.The Frölicher spectral sequence of X degenerates at page 2, with E p,q 2 = H p dR (K, C) ⊗ H q (Γ, C).We will give two proofs of this theorem, each one highlighting somewhat different aspects.Both will use the following computation of Dolbeault cohomology by Akhiezer.Note that the left-multiplication of G on X induces a holomorphic representation of G on the Dolbeault cohomology H p,q ∂ (X).
Theorem 12 (Akhiezer [1]).There is a G-module isomorphism where Λ p g 1,0 is a G-module induced by the adjoint representation and H q (Γ, C) is a trivial G-module.
Proof of Theorem 11 via 4 spectral sequences.Claim 1.The Hochschild-Serre spectral sequence for k ⊂ g degenerates at the second page.We notice that H * (g Lemma 10.Let E p,q r(HS) be the Hochschild-Serre spectral sequence for k ⊂ g.We have Thus the claim follows.
Claim 2. The Serre spectral sequence for the fiber bundle X → Z degenerates at the second page.Let E p,q r(S) be the Serre spectral sequence for the fiber bundle X → Z. Since X → Z is a principal K-bundle, we have E p,q 2(S) = H p dR (Z) ⊗ H q dR (K).By the multiplicative structure on the spectral sequence, we have d p,q 2 = Id H p dR (Z) ⊗ d 0,q 2 .We notice that the inclusion i : Λg * ⊂ A * X induces a morphism i p,q r : E p,q r(HS) → E p,q r(S) .By the averaging, we have a map µ : A * X → Λg * such that µ • i = Id.This implies that i p,q r : E p,q r(HS) → E p,q r(S) is injective.By E 0,q r(HS) ∼ = H q (k) ∼ = H q dR (K), we can say that i 0,q 2 : E 0,q r(HS) → E 0,q r(S) is an isomorphism.By the first claim, we have d 0,q 2 = 0 on E 0,q 2(HS) and hence d 0,q 2 = 0 on E 0,q 2(S) .This implies d p,q 2 = 0 on E p,q 2(S) .By the same argument, we can say d p,q r = 0 on E p,q r(S) for any r ≥ 2. Claim 3. The Frölicher spectral sequence of X degenerates at the second page.Let E p,q r(F S) be the Serre spectral sequence for the fiber bundle X → Z.By Akhiezer's theorem, we have E p,q 1(F S) = H p,q ∂ (X) = Λ p g 1,0 ⊗ H q (Γ, C).The differential d p,q 1 on E p,q 1(F S) = H p,q ∂ (X) is induced by the differential ∂ and we can easily check that it is determined by the g 1,0 -module structure of H 0,q ∂ (X) = H q (Γ, C).Since H q (Γ, C) is a trivial G-module, g 1,0 acts trivially on H 0,q (X) and we can say that the differential ∂ induces a trivial differential on H q (Γ, C).Thus we have E p,q 2(F S) ∼ = H p (g 1,0 ) ⊗ H q (Γ, C).By the above argument, we have ).By the E 2 -degeneration of E p,q r(S) , for any r ∈ Z we have p+q=r dim E p,q 2(F S) = p+q=r dim E p,q 2(S) = dim H r dR (X) and hence the claim follows.
Proof of Theorem 11 via lattice-cohomology subcomplex of A X .The lattice cohomology H(Γ, C) can be computed via the complex A Γ Y of Γ-invariant forms on Y [5, Ch.VII].By pulling forms back to G then pushing forward to X, we obtain a map of (simple) complexes We will also consider the projection map pr : A X → (A 0, • X , ∂) which is a map of complexes and induces the 'edge maps' Admitting the claim, the edge maps are surjective, hence there can be no differentials starting at E 0,q r , for any q and r ≥ 1.Thus, by the Leibniz-rule, any possible nonzero differential has to live on E 2 and be of the form Id ⊗d g 1,0 .
In order to prove the claim, we identify (c.f.[5, Ch.VII]): , where the action of K is induced by right multiplication on X on the right factor and the adjoint action on the left factor, ).Under these identifications, consider the map σ : A → A X induced by the inclusion (g/k) ∨ C → g ∨ C .Since the composition g 0,1 → g C → (g/k) C is an isomorphism, so is pr •σ.On the other hand, pr •σ is an equivariant map, therefore it identifies the K-invariants on the left (i.e.A Γ Y ) with the K-invariants on the right.But by averaging over K, one sees that the complex of K-invariants is a direct summand in A 0, •.Therefore, the edge map H q (Γ, C) → E 0,q 1 induced by pr •σ has to be injective.Since by Akhiezer's result both source and target have the same dimension, the claim follows.
We consider the canonical injection H k (g; k, C) ֒→ H k (Γ, C) associated with the locally symmetric space Z := Γ\G/K.In view of Theorem 12 it can be identified with the map induced by the inclusion of left-invariant forms H 0,k ∂ (Λ g ) ֒→ H 0,k ∂ (X).
Corollary 13.X is a page-1-∂ ∂ manifold if and only if H k (g, k, C) ∼ = H k (Γ, C) for any k ∈ Z.
Proof.Suppose X is a page-1-∂ ∂ manifold.By Remark 3, we have dim E p,q 2 = dim E q,p 2 .By Theorem 11, this implies dim H k dR (K) = dim H k (Γ) for any k ∈ Z.As we saw in the first proof of Theorem 11, we have H k (g; k) ∼ = H k dR (K).Thus we have dim H k (g; k, C) = dim H k (Γ, C) and so the injection H k (g; k, C) ֒→ H k (Γ, C) is an isomorphism.
Hence the injection H p,q ∂ (Λ g ) ֒→ H p,q ∂ (X) is an isomorphism and so X is a page-1-∂ ∂ by Proposition 8.