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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 ...

Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [5] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

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Abstract. A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived.A classical result of I. M. Singer [11] states that a Riemannian manifold is locally homogeneous if and only if its Riemannian curvature tensor together with its covariant derivatives up to some index k + 1 are independent of the point (the integer k is called the Singer invariant). More precisely, Theorem 1 (Singer [11]). Let M be a Riemannian manifold. Then M is a locally homogeneous if and only if for any p, q ∈ M there is a linear isometryfor any s ≤ k + 1.An alternate proof with a more direct approach was given in [8].Out of the curvature tensor and its covariant derivatives one can construct scalar invariants, like for instance the scalar curvature. In general, any polynomial function in the components of the curvature tensor and its covariant derivatives which does not depend on the choice of the orthonormal basis at the tangent space of each point is a scalar Weyl invariant or a scalar curvature invariant. By the Weyl theory of invariants, a scalar Weyl invariant is a linear combination of complete traces of tensors