Fr¨olicher spectral sequence and Hodge structures on the cohomology of complex parallelisable manifolds Hisashi Kasuya, Jonas Stelzig November 17, 2020 Abstract For complex parallelisable manifolds Γ\G, with G a solvable or semisimple complex Lie group, the Fr¨olicher 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. 1 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 Fr¨olicher spectral sequence of a compact complex paral- arXiv:2011.07905v1 [math.AG] 16 Nov 2020 lelisable manifold X = Γ\G. If the Fr¨olicher 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 Fr¨olicher spectral sequence of X degenerates at the first page if and only if G is abelian. Hence, unlike com- pact K¨ahler 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 Fr¨olicher spectral sequence degenerates at the second page and the Hodge 1 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 = 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 Fr¨olicher 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 ana- logue 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 Fr¨olicher 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 dou- ble 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 Fr¨olicher spectral sequence.