Abstract. This paper classifies Hermitian structures on 6-dimensional nilmanifolds M = Γ\G for which the fundamental 2-form is ∂∂-closed, a condition that is shown to depend only on the underlying complex structure J of M . The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limaçon-shaped curve in the complex plane. Related theory is used to provide examples of various types of Ricci-flat structures. Mathematics Subject Classification (2000). 53C55; 32G05, 17B30, 81T30.

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 ...

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structure, thus answering a question in [16]. We also prove that the existence of HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure with restricted holonomy of its Bismut connection in SU(n), thus providing a counter-example of the conjecture in [18]. Again we prove that such property is not stable under small deformations.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form ∂∂-closed, i.e., to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if M is a nilmanifold (not a torus) endowed with an invariant complex structure J, then (M, J) does not admit any symplectic form taming J. Moreover, we show that if a nilmanifold M endowed with an invariant complex structure J admits an SKT metric, then M is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application we construct a holomorphic family (M, Ja) of compact complex manifolds such that (M, Ja) satisfies the ∂∂lemma and admits a balanced metric for any a = 0, but the central limit neither satisfies the ∂∂-lemma nor admits balanced metrics.