Fr\" Olicher Spectral Sequence and Hodge Structures on The

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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 = SL2(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. Theorem A. If G is solvable, X is a page-1-∂∂¯ manifold. 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 = SL2(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 Hk(g; k, C) ֒→ Hk(Γ, C) where Hk(g, k, C) is the relative Lie algebra cohomology and Hk(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 Fr¨olicher spectral sequence of X degenerates at the second page. 2. X is a page-1-∂∂¯ manifold if and only if Hk(g, k, C) =∼ Hk(Γ, C) for any k ∈ Z. In view of Theorems A and B, it appears natural to make the following: Conjecture. For any compact complex parallelisable manifold X, the Fr¨olicher spectral sequence degenerates at the second page. Apart from the results in this article, indication that this might hold is given 0,1 2,0 by [11], where it is shown that d2 : E2 (X) → E2 (X) always vanishes.

2 Preliminaries 2.1 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 := Ap,q · p+q=· L 2 the associated total complex with differential d := ∂ + ∂¯. Associated with the r p,q filtration F A = p≥r A we get an induced filtration on the total cohomology HdR· (A) := H·(TotLA), still denoted by F , and the spectral sequence p,q p,q p+q E1 (A)= H∂¯ (A)=⇒ (HdR (A), F ).

r p,q Analogously, for the filtration F A = q≥r A we obtain a second spectral sequence converging to HdR(A), whichL we denote by E¯r(A). It has the spaces p,q H∂ (A) on its first page. The Bott-Chern cohomology is defined as ker ∂ : Ap,q → Ap+1,q ∩ ker ∂¯ : Ap,q → Ap,q+1 Hp,q (A) := BC im ∂∂¯ : Ap−1,q−1 → Ap,q and the Aeppli cohomology as

ker ∂∂¯ : Ap−1,q−1 → Ap,q ) Hp,q(A) := . A im (∂ : Ap−1,q→ Ap,q) + im ∂¯ : Ap,q−1 → Ap,q All double complexes which we will consider will have finite dimensional p,q p,q cohomology, i.e. that the spaces H∂¯ (A) and H∂ (A) are finite dimensional for Z p,q ¯p,q p,q p,q k all p, q ∈ (this also implies that Er (A), Er (A),HBC (A),HA (A),HdR(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. Ap,q = Aq,p and ∂a¯ = ∂a¯ . In this case the second spectral sequence is determined by the first and can be ignored. p,q p,q p+q We denote by H (A) := im(HBC (A) −→ HdR (A)). We have equalities p,q p,q p ¯q p+q H (A)= {[α] ∈ H·(TotA) : α ∈ A } = (F ∩ F )HdR (A). 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: 1. ker ∂ ∩ ker ∂¯ ∩ imd = im∂∂¯ 2. Both spectral sequences degenerate at page 1 and for all k ∈ Z, the filtra- ¯ k tions F and F on HdR(A) induce a pure Hodge structure of degree k, i.e. k p,q HdR(A)= p+q=k H (A). L p,q 3. Every class in H∂¯ (A) admits a d-closed pure-type representative and p,q p,q the map H∂¯ (A) −→ H (A) sending a class via such a representative induces a well-defined isomorphism

p,q H∂¯ (A) → HdR· (A) p,qM∈Z

p,q (and analogously for H∂ (A)). 4. A is a direct sum of complexes of the following types

3 (a) ‘squares’: complexes with a single pure-bidegree generator a, s.t. ∂∂a¯ 6=0 h∂a¯ i h∂∂a¯ i

hai h∂ai

(b) ‘dots’: complexes concentrated in a single bidegree, with all differen- tials being zero. hai The main interest of this paper is following analogue of this property, intro- duced 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 ¯ k filtrations F and F on HdR(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. p,q p,q r p,q Define h# (A) = dim H# (A) and h#(A) = r=p+q h# (A) for each # = ∂, ∂,BC,A¯ . P Proposition 2 ([13]). Let A be a bounded double complex with finite-dimensional cohomology. The following assertions are equivalent: 1. A is page-1-∂∂¯: p,q 2. Every class in E2 (A) admits a d-closed pure-type representative and the p,q p,q map E2 (A) → H (A) sending a class via such a representative induces a well-defined linear isomorphism

p,q E2 (A) → HdR· (A) M and analogously for E¯2. Z r r r r 3. for every r ∈ the equality hA(A)+ hBC (A)= h∂¯(A)+ h∂ (A) holds. 4. 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.

h∂a¯ i hai h∂ai

hai

There is also a characterisation in terms of suitable exactness properties (c.f. [14] for details).

4 Remark 3. Note that for complexes with real structure property 2 implies the p,q q,p symmetry dim E2 = dim E2 . , ¯ Let X be a complex manifold and AX = (AX··, ∂, ∂) the Dolbeault double complex, i.e. the double complex of C-valued differential forms. We de- p,q p,q note Er (X) = Er (AX ) and call it the Frölicher spectral sequence. In C this case the total cohomology is the de Rham cohomology HdR· (X, ) of X, ¯ p,q the ∂-cohomology H∂¯ (AX ) is the Dolbeault cohomology of X. We denote p,q p,q p,q p,q r r H# (X) = H# (AX ), h# (X) = h# (AX ) and h#(X) = h#(AX ) for each #= ∂, ∂,BC,A¯ . Definition 4. Let X be a compact complex manifold. X is called a page-1-∂∂¯- manifold if the Dolbeault double complex AX is page-1-∂∂¯. 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 → AX of double complexes, such that for all p, q ∈ Z the p,q p,q ¯ induced map H# (C) → H# (X) is an isomorphism for # = ∂, ∂, then it is also an isomorphism for each # = BC,A. Moreover, if C has the page-1-∂∂¯- property, then also X is a page-1-∂∂¯-manifold.

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.

Lemma 6 ([13, Lemma 4.6.]). If A = (A·, d1), B = (B·, d2) are (simple) complexes, then the double complex A⊗B = (A·⊗B·, d1⊗IdB, IdA⊗d2) is page- ¯ p,q p,q p q 1-∂∂. The spaces E2 (A⊗B)= H (A⊗B) are identiﬁed with H (A)⊗H (B).

2.2 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 gC = g1,0 ⊕ g0,1 into i and −i Eigenspaces and we denote ∨ 1,0 0,1 ∨ by gC = g ⊕ g the splitting of the dual space. Denote by Λg :=Λ·gC ⊆ AG the space of left invariant forms on G. The restriction of the exterior diﬀerential + 1,0 makes it into a sub-double complex of AG. Denote by Λg = (Λ·g , ∂) and − 0,1 ¯ Λg = (Λ·g , ∂) 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 Λg is the tensor product of the simple complexes Λg − ¯ and Λg , 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

5 by the left action of Γ. Since Λg consists of left-invariant forms, it can also be considered as a sub-double complex of AX . It carries the induced real structure. Hence, by Lemma 7 and Corollary 5 we have:

Proposition 8. If the inclusion i : Λg → AX induces a cohomology isomor- p,q ∼ p,q ¯ phism H∂¯ (Λg) = H∂¯ (X), then X is a page-1-∂∂-manifold.

In [16], Sakane proved that if G is nilpotent, then the inclusion i : Λg → p,q ∼ p,q AX induces a cohomology isomorphism H∂¯ (Λg) = H∂¯ (X). Hence, if G is nilpotent, then X is a page-1-∂∂¯-manifold. If G is not nilpotent the inclusion i : Λg → AX does not induce an isomor- p,q ∼ p,q phism H∂¯ (Λg) = H∂¯ (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 : AX → Λg such that µ ◦ i = Id (see the proof of [4, Theorem 7]). Thus, one obtains a direct sum decomposition AX = Λg ⊕ AX /Λg and a corresponding decomposition on all cohomologies considered. In particular, the cohomology groups of Λg inject into those of X.

3 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 simply- connected complex nilpotent subgroup C ⊆ G such that G = C · N. Associated with C, we deﬁne the diagonalizable representation

G = C · N ∋ c · n 7→ (Adc)s ∈ Aut(g1,0) where (Adc)s is the semi-simple part of the Jordan decomposition of the adjoint operator (Adc) ∈ Aut(g1,0). Denote this representation by Ads : G → Aut(g1,0). We have a basis X1,...,Xn of g1,0 such that,

Ads(g) = diag (α1(g),...,αn(g)) for some characters α1,...,αn of G. Let BΓ· be deﬁned as

α¯I α¯I B· := x¯I I ⊆{1,...,n} such that =1 , Γ α α I I Γ

(where we shorten, e.g. αI := αi1 ····· αik for a multi-index I = (i1,...,ik)). 0, ∼ 0, Then the inclusion BΓ· ⊂ AX· induces an isomorphism H·(BΓ·) = H ·(X) ([9]). Consider the following subcomplex of AX :

1, 2 1 1,0 2 1 2 0,1 C· · := Λ· g ⊗C BΓ· + BΓ· ⊗C Λ· g .

1, 2 , Then C· · is a double complex with real structure and the inclusion CΓ·· ⊂ , p,q 1, 2 ∼ p,q AX·· induces an isomorphism H∂¯ (CΓ· · ) = H∂¯ (X) ([3, Section 2.6]). By

6 Corollary 5), it suﬃces to show that C·1,·2 is page-1-∂∂¯ in order to prove Theorem A. 1,0 Take the weight decomposition Λ·g = λ Vλ for the representation Ads : 0,1 0,1 G → Aut(g1,0). We have Λ·g = λ Vλ¯ .L (We must distinguish V1¯ ⊂ Λ·g 1,0 from V1 ⊂ Λ·g for the trivial representationL 1). Then we can write

1 1,0 2 α Λ· g ⊗C B· = Vλ ⊗  Vα¯  Γ α¯ Mλ αM ( α¯ )⌊Γ=1    and

1 2 0,1 α¯ B· ⊗C Λ· g =  Vα ⊗ Vλ¯ . Γ α Mλ αM ( α¯ )⌊Γ=1    We have

1 1,0 2 1 2 0,1 1 Λ· g ⊗C B· ∩ B· ⊗C Λ· g = αV 1 ⊗ Vα¯ Γ Γ α α¯ αM ( α¯ )⌊Γ=1 and hence

1, 2 1 1 1 C· · =  αVλ ⊗ Vα¯ ⊕ αV 1 ⊗ Vα¯ ⊕ Vα ⊗ αV¯ λ¯  . α¯ α α¯ α αM λM6= 1 λM6= 1  ( α¯ )⌊Γ=1 α α   ¯ By Ads(g) ∈ Aut(g1,0), we have d = ∂ : Vλ → Vλ and also d = ∂ : Vλ¯ → Vλ¯ . −1 Since we have α dα ∈ V1 for any holomorphic character α, we have d = ∂ : ¯ αVλ → αVλ and also d = ∂ : αV¯ λ¯ → αV¯ λ¯ 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.

3.1 Variant for Complex Solvmanifolds of Splitting Type In this subsection only, G will not be a complex Lie group, but we assume that n G is the semi-direct product C ⋉φ N so that: 1. N is a connected simply-connected 2m-dimensional nilpotent Lie group endowed with an N-bi-invariant complex structure JN ; (denote the Lie algebras of Cn and N by a and, respectively, n;) 2. for any t ∈ Cn, it holds that φ(t) ∈ Aut(N) is a holomorphic automor- phism of N with respect to JN ; 3. φ induces a semi-simple action on n;

4. G has a lattice Γ; (then Γ can be written as Γ = ΓCn ⋉φ ΓN such that n ΓCn and ΓN are lattices of C and, respectively, N, and, for any t ∈ ΓCn , it holds φ(t)(ΓN ) ⊆ ΓN .)

7 Consider the solvmanifold X = Γ\G. This satisﬁes [8, Assumption 1.1] ([3, Assumption 2.11]). Theorem 9. X is a page-1-∂∂¯ manifold. This result is a generalization of the observation on completely solvable Naka- mura manifolds in [13, Corollary 3.3].

n Proof. Consider the standard basis {X1,...,Xn} of C . Consider the decomposition n ⊗R C = n1,0 ⊕ n0,1 induced by JN . By the condition 2, this decomposition is a direct sum of Cn-modules. By the condition 3, we have a basis n ∗ {Y1,...,Ym} of n1,0 and characters α1,...,αm ∈ Hom(C ; C ) such that the induced action φ on n1,0 is represented by

n C ∋ t 7→ φ(t) = diag (α1(t),...,αm(t)) ∈ Aut(n1,0) .

For any j ∈ {1,...,m}, since Yj is an N-left-invariant (1, 0)-vector ﬁeld on n N, the (1, 0)-vector ﬁeld αj Yj on C ⋉φ N is G-left-invariant. Consider the Lie algebra g of G and the decomposition gC := g ⊗R C = g1,0 ⊕ g0,1 induced by J. Hence we have a basis {X1,...,Xn, α1Y1,...,αmYm} of g1,0, and let −1 −1 1,0 x1,...,xn, α1 y1,...,αm ym be its dual basis of g . Then we have p −1 −1 p,q Λ x1,...,xn, α1 y1,...,αm ym Λ gC = q −1 −1 . ⊗Λ x¯1,..., x¯n, α¯1 y¯1,..., α¯m y¯m

n ∗ For any j ∈{1,...,m}, there exist unique unitary characters βj ∈ Hom(C ; C ) Cn C∗ Cn −1 −1 and γj ∈ Hom( ; ) on such that αj βj andα ¯j γj are holomorphic [8, Lemma 2.2]. , , Deﬁne the diﬀerential bi-graded sub-algebra BΓ·· ⊆ AX·· as

p,q −1 −1 B := C xI ∧ α βJ yJ ∧ x¯K ∧ α¯ γL y¯L . Γ J L |I|+|J|=p and |K|+|L|=q such that (βJ γL)⌊Γ=1

Deﬁne , , , CΓ·· := BΓ·· + BΓ··. , , Then CΓ·· is a double complex with real structure and the inclusion CΓ·· ⊂ , p,q , ∼ p,q , , ∼ , AX·· induces an isomorphism H∂¯ (CΓ··) = H∂¯ (X) and HBC··(CΓ··) = HBC··(X) , ¯ ([3, Section 2.5]). Again, it suﬃces to show that CΓ·· is page-1-∂∂. 1,0 1,0 0,1 0,1 Take the weight decomposition Λ·n = α Vα and Λ·n = α¯ Vα¯ for the Cn-action. Then we have L L , , Cn −1 1,0 ¯ −1 0,1 BΓ·· = Λ·· ⊗ (αβ Vα ) ⊗ (λγ Vλ¯ ) (βγM)⌊Γ=1

8 where β ∈ Hom(Cn; C∗) and γ ∈ Hom(Cn; C∗) are unitary characters on Cn such that αβ−1 and λγ¯ −1 are holomorphic. We have

, Cn −1 ¯ −1 1,0 0,1 CΓ·· = Λ ⊗ (αβ λγ Vα ) ⊗ Vλ¯ (βγ)⌊ΓM=1,αλ¯6=βγ Cn 1,0 −1 ¯ −1 0,1 ⊕ Λ ⊗ Vλ ⊗ αβ λγ Vα¯ (βγ)⌊ΓM=1,αλ¯6=βγ Cn 1,0 0,1 ⊕ Λ ⊗ Vα ⊗ Vλ¯ . (αλ¯M)⌊Γ=1

1,0 1,0 ¯ 0,1 0,1 Since JN is bi-invariant, we have d = ∂ : Vα → Vα and d = ∂ : Vα¯ → Vα¯ . Since αβ−1λγ¯ −1 is holomorphic, we have

−1 ¯ −1 1,0 1,0Cn −1 ¯ −1 1,0 d = ∂ : αβ λγ Vα → Λ ⊗ (αβ λγ Vα ) and ¯ −1 ¯ −1 0,1 0,1Cn −1 ¯ −1 0,1 d = ∂ : αβ λγ Vα¯ → Λ ⊗ (αβ λγ Vα¯ ). Thus each each factor of the direct sum of C·,· is the tensor product of simple complexes, hence it is a page-1-∂∂¯-complex.

4 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:

Lemma 10. There is a canonical isomorphism H(g1,0)= HdR(K, C).

Proof. This is a consequence of the identification of complex Lie algebras g1,0 =∼ 1,0 ∼ ∨ g = k⊕Jk = kC (whence the identification Λ·g = Λ·kC ) 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 p,q p C q C E2 = HdR(K, ) ⊗ H (Γ, ). 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 p,q holomorphic representation of G on the Dolbeault cohomology H∂¯ (X).

9 Theorem 12 (Akhiezer [1]). There is a G-module isomorphism p,q p 1,0 q C H∂¯ (X)=Λ g ⊗ H (Γ, ) where Λpg1,0 is a G-module induced by the adjoint representation and Hq(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 (gC)= H (g1,0) ⊗ H (g0,1) =∼ H (kC) ⊗ H (kC) p,q by Lemma 10. Let Er(HS) be the Hochschild-Serre spectral sequence for k ⊂ p,q p q p k q C C ∗ C g. We have E2(HS) = H (g; k, ) ⊗ H (k ) = (Λ (Jk)C) ⊗ H (k ) by the decomposition g = k + Jk with [Jk,Jk] ⊂ k and [k,Jk] ⊂ Jk. Since J gives a ∗ p ∗ k p ∗ k k-module isomorphism k =∼ Jk, we have H (kC) =∼ (Λ (k)C) =∼ (Λ (Jk)C) . Thus the claim follows. Claim 2. The Serre spectral sequence for the fiber bundle X → Z degenerates p,q at the second page. Let Er(S) be the Serre spectral sequence for the fiber bundle p,q p X → Z. Since X → Z is a principal K-bundle, we have E2(S) = HdR(Z) ⊗ q HdR(K). By the multiplicative structure on the spectral sequence, we have p,q 0,q ∗ ∗ d = Id p ⊗ d . We notice that the inclusion i : Λg ⊂ A induces a 2 HdR(Z) 2 X p,q p,q p,q ∗ morphism ir : Er(HS) → Er(S). By the averaging, we have a map µ : AX → ∗ p,q p,q p,q Λg such that µ ◦ i = Id. This implies that ir : Er(HS) → Er(S) is injective. 0,q ∼ q ∼ q 0,q 0,q 0,q By Er(HS) = H (k) = HdR(K), we can say that i2 : Er(HS) → Er(S) is an 0,q 0,q 0,q isomorphism. By the first claim, we have d2 =0on E2(HS) and hence d2 =0 0,q p,q p,q on E2(S). This implies d2 = 0 on E2(S). By the same argument, we can say p,q p,q dr =0 on Er(S) for any r ≥ 2. Claim 3. The Frölicher spectral sequence of X degenerates at the second p,q page. Let Er(FS) be the Serre spectral sequence for the fiber bundle X → Z. p,q p,q p 1,0 q C By Akhiezer’s theorem, we have E1(FS) = H∂¯ (X)=Λ g ⊗ H (Γ, ). The p,q p,q p,q differential d1 on E1(FS) = H∂¯ (X) is induced by the differential ∂ and we can g 0,q easily check that it is determined by the 1,0-module structure of H∂¯ (X) = q q 0,q H (Γ, C). Since H (Γ, C) is a trivial G-module, g1,0 acts trivially on H (X) and we can say that the differential ∂ induces a trivial differential on Hq(Γ, C). p,q ∼ p q C Thus we have E2(FS) = H (g1,0) ⊗ H (Γ, ). By the above argument, we have p,q p q p q p q C C C ∼ C C ∼ C E2(S) ⊗ = H (Z, ) ⊗ H (K, ) = H (Γ, ) ⊗ H (k ) = H (Γ, ) ⊗ H (g1,0). p,q Z p,q By the E2-degeneration of Er(S), for any r ∈ we have p+q=r dim E2(FS) = p,q r p+q=r dim E2(S) = dim HdR(X) and hence the claim follows.P P Proof of Theorem 11 via lattice-cohomology subcomplex of AX . The lattice co- C Γ homology H(Γ, ) can be computed via the complex AY 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

Γ σ : AY −→ AX

10 We will also consider the projection map 0, ¯ pr : AX → (AX·, ∂) which is a map of complexes and induces the ‘edge maps’

q q HdR(X) → H∂¯(X) q C ∼ 0,q q C Claim: The composition pr ◦σ induces isomorphisms H (Γ, ) = E1 = H (Γ, ).

Admitting the claim, the edge maps are surjective, hence there can be no 0,q diﬀerentials starting at Er , for any q and r ≥ 1. Thus, by the Leibniz-rule, any possible nonzero diﬀerential has to live on E2 and be of the form Id ⊗dg1,0 .

In order to prove the claim, we identify (c.f. [5, Ch. VII]): Γ ∨ ∞ C 1. AY with the K-invariants of A := Λ(g/k)C ⊗C (X, ), where the action of K is induced by right multiplication on X on the right factor and the adjoint action on the left factor,

∞ 2. AX with Λg ⊗C (X, C), 0, − ∞ C 3. AX· with Λg ⊗C (X, ).

Under these identifications, consider the mapσ ˜ : A → AX induced by the ∨ ∨ inclusion (g/k)C → gC. Since the composition g0,1 → gC → (g/k)C is an isomorphism, so is pr ◦σ˜. On the other hand, pr ◦σ˜ is an equivariant map, therefore Γ it identifies the K-invariants on the left (i.e. AY ) with the K-invariants on the right. But by averaging over K, one sees that the complex of K-invariants is 0, q C 0,q a direct summand in A ·. Therefore, the edge map H (Γ, ) → E1 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 Hk(g; k, C) ֒→ Hk(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 0,k 0,k .(H∂¯ (Λg) ֒→ H∂¯ (X Corollary 13. X is a page-1-∂∂¯ manifold if and only if Hk(g, k, C) =∼ Hk(Γ, C) for any k ∈ Z. ¯ p,q Proof. Suppose X is a page-1-∂∂ manifold. By Remark 3, we have dim E2 = q,p k k Z dim E2 . By Theorem 11, this implies dim HdR(K) = dim H (Γ) for any k ∈ . k ∼ k As we saw in the first proof of Theorem 11, we have H (g; k) = HdR(K). Thus →֒ (we have dim Hk(g; k, C) = dim Hk(Γ, C) and so the injection Hk(g; k, C Hk(Γ, C) is an isomorphism. We assume Hk(g, k, C) =∼ Hk(Γ, C) for any k ∈ Z. Then by Theorem 12 and Lemma 10, we have p,q pg1,0 q g p,q dim H∂¯ (X) = dim Λ ⊗ H ( 1,0) = dim H∂¯ (Λg).

11 p,q p,q Hence the injection H∂¯ (Λg) ֒→ H∂¯ (X) is an isomorphism and so X is a page-1-∂∂¯ by Proposition 8.

Remark 14. For G = SL2(C) with K = SU(2), b1(K) = b2(K)=0 but for any r ∈ N, there exist lattices Γ ⊆ G such that b1(Γ) = b2(Γ) = r (see [20, §B], [7, 6.2.]). Note that by a theorem of Raghunathan [15], b1(Γ) = 0 as soon as G has no SL2(C)-factor. Moreover, for k < rkRg, the injection .([Hk(g; k, C) ֒→ Hk(Γ, C) is an isomorphism(see [5, VII, Corollary 4.4

References

[1] Akhiezer, D. Group actions on the dolbeault cohomology of homogeneous manifolds. Math. Z. 226, no. 4 (1997), 607–621. [2] Angella, D. The cohomologies of the iwasawa manifold and of its small deformations. J. Geom. Anal. 23, no. 3 (2013), 1355–1378. [3] Angella, D., and Kasuya, H. Bott-chern cohomology of solvmanifolds. Ann. Global Anal. Geom. 52, no. 4 (2017), 1571–1574. [4] Belgun, F. A. On the metric structure of non-kähler complex surfaces. Math. Ann. 317, no. 1 (2000), 1–40. [5] Borel, A., and Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition., vol. 67 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000. [6] Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D. Real homotopy theory of kähler manifolds. Invent. Math. 29, 3 (1975), 245–274. [7] Ghys, E. Déformations des structures complexes sur les espaces homogènes de SL(2, C). J. Reine Angew. Math. 468 (1995), 113–138. [8] Kasuya, H. Techniques of computations of dolbeault cohomology of solvmanifolds. Math. Z. 447 (2013), 437–447. [9] Kasuya, H. de rham and dolbeault cohomology of solvmanifolds with local systems. Math. Res. Lett. 21, no. 4 (2014), 781–805. [10] Kasuya, H. The frölicher spectral sequence of certain solvmanifolds. J. Geom. Anal. 25, no. 1 (2015), 317–328. [11] Lescure, F. Annulation d’une deuxième différentielle. J. reine angew. Math. 527 (2000), 37–68. [12] Milnor, J. Curvatures of left invariant metrics on lie groups. Advances in Math. 21, no.3 (1976), 293–329.

12 [13] Popovici, D., Stelzig, J. and Ugarte, L. Higher-page Hodge theory of Compact Complex Manifolds. arXiv:2001.02313 [14] Popovici, D., Stelzig, J. and Ugarte, L. Higher-Page Bott-Chern and Aeppli Cohomologies and Applications. arXiv:2007.03320 [15] Raghunathan, M. S. Vanishing theorems for cohomology groups associated to discrete subgroups of semisimple lie groups. Osaka J. Math. 3, 2 (1966), 243–256. [16] Sakane, Y. On compact complex parallelisable solvmanifolds. Osaka J. Math. 13 (1976), 187–212. [17] Schmid, W. Vanishing theorems for lie algebra cohomology and the cohomology of discrete subgroups of semisimple lie groups. Adv. in Math. 41, no. 1 (1981), 78–113. [18] Stelzig, J. On the Structure of Double Complexes arXiv:1812.00865 [19] Wang, H.-C. Complex parallelisable manifolds. Proc. Amer. Math. Soc. 5 (1954), 771–776. [20] Winkelmann, J. Complex-analytic geometry of complex paralleliz- able manifolds. habilitationsschrift. Heft 13 der Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik an der Ruhr- Universit¨at Bochum (1995).