Dolbeault cohomology of compact nilmanifolds

Abstract

Abstract. Let M = G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space C(g) of invariant complex structures on M , the Dolbeault cohomology of M is isomorphic to the one of the differential bigraded algebra associated to the complexification g C of the Lie algebra of G. To obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures. IntroductionLet M be a compact nilmanifold of real dimension 2n. It follows from a result of Mal'čev [16] that M = G/Γ where G is a simply connected (s + 1)-step nilpotent Lie group admitting a basis of left invariant 1-forms for which the coefficients in the structure equations are rational numbers, and Γ is a lattice in G of maximal rank (i.e., a discrete uniform subgroup, cf. [23]). We will let Γ act on G on the left. It is well known that such a lattice Γ exists in G if and only if the Lie algebra g of G has a rational structure, i.e. if there exists a rational Lie subalgebra g Q such that g ∼ = g Q ⊗ R.The de Rham cohomology of a compact nilmanifold can be computed by means of the cohomology of the Lie algebra of the corresponding nilpotent Lie group (Nomizu's Theorem [21]).We assume that M has an invariant complex structure J, that is to say that J comes from a (left invariant) complex structure J on g. Our aim is to relate the Dolbeault cohomology of M with the cohomology ring H * , * ∂ (g C ) of the differential bigraded algebra Λ * , * (g C ) * , associated to g C with respectto the operator ∂ in the canonical decomposition d = ∂ + ∂ on Λ * , * (g C ) * . The study of the Dolbeault cohomology of nilmanifolds with an invariant complex structure is motivated by the fact that the latter provided the first known examples of compact symplectic manifolds which do not admit any Kähler structure [1,5,28].Since there exists a natural mapwhich is always injective (cf. Lemma 7), the problem we will study is to see for which complex structure J on M the above map gives an isomorphism1991 Mathematics Subject Classification. 53C30, 53C35.Research partially supported by MURST and CNR of Italy. Note that H * , * ∂ (g C ) can be identified with the cohomology of the Dolbeault complex of the forms on G which are invariant by the left action of G (we shall call them briefly G-invariant forms) and H * , * ∂ (M ) with the cohomology of the Dolbeault complex of Γ-invariant forms on G. We shall use these identifications throughout this note.Our main result is the following Theorem A The isomorphism (1) holds on an open set of any connected component of the moduli space C(g) of invariant complex structures on M .To obtain Theorem A we first consider the case of complex structures J which ...

Erratum to: The Frölicher spectral sequence can be arbitrarily non-degenerate

“…By the results of [2] the inclusion of left-invariant forms into the double complex (A p,q (X n ), ∂,∂) induces an isomorphism on the E 1 -terms of the respective spectral sequences. Thus for our purpose we may work with left-invariant forms only, that is, start with the E 0 -term…”

Stability of Abelian Complex Structures

“…As in [3] [4] [5], we need to identify Dolbeault and Lie algebra cohomology, in the spirit of [6]. To this aim we construct a chain of fibrations with tori as fibres which are associated to the ascending series considered above.…”

Almost parallel structures

“…Some tools are available to compute the Dolbeault cohomology of an invariant complex structure on N/Γ (Cordero et al, 2000;Console and Fino, 2001;Pedersen and Poon, 2001), but not in the most general situation. It is therefore natural to study the convergence of the Frölicher spectral sequence E Semisimple examples, but in somewhat higher dimensions, were provided by (Pittie, 1989).…”