Luis A. Cordero

Symplectic manifolds with no kahler structure

On decrit plusieurs familles de varietes symplectiques compactes qui generalisent la variete de Kodaira-Thurston

Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if Γ\G is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology H∗,∗ ∂̄ (Γ\G) is canonically isomorphic to the ∂̄–cohomology H∗,∗ ∂̄ (gC) of the bigraded complex (Λ∗,∗(gC)∗, ∂̄) of complex valued left invariant differential forms on the nilpotent Lie group G.

A general description of the terms in the Frölicher spectral sequence

An explicit description of the Frolicher spectral sequence is given that is useful for calculations for compact nilmanifolds. We use the description to exhibit examples of compact complex nilmanifolds of (complex) dimension 3 for which the Frolicher spectral sequence does not degenerate at the second term.

Almost Complex Poisson Manifolds

In this paper we consider complex Poisson manifolds and extendthe concept of complex Poisson structure, due to Lichnerowicz to themore general concept of almost complex Poisson structures. Examples ofsuch structures and the associated generalized foliation are given.Moreover, some properties of the complex symplectic structures as wellas of the holomorphic complex Poisson structures are studied.

Lifts of tensor fields to the frame bundle

LetM be a differentiable manifold,T M its tangent bundle andFM its frame bundle. The theory of lifts toT M of tensor fields onM has been extensively studied by many authors. In this paper, a similar theory for the frame bundle is developed by introducing the complete, horizontal and diagonal lifts toFM of tensor fields onM, with the aim of making this study as closely comparable with that forT M as possible.

Frölicher Spectral Sequence of Compact Nilmanifolds with Nilpotent Complex Structure

We consider a special class of compact complex nilmanifolds; namely, compact nilmanifolds with nilpotent complex structure [10]. For such a manifold Г\G we prove that the terms E r (Г\G) in the Frolicher spectral sequence are canonically isomorphic to the terms E r (g ℂ) in the spectral sequence at the Lie algebra level g of G. Moreover, we show that the bidifferential bigraded algebra (A*,*(g ℂ), ∂,∂) is a model for the double complex (⋀*,*(Г\G), ∂, ∂). We also construct new examples of compact complex manifolds of dimension 3 (the lowest possible dimension) with E 2 ≇ E∞,, and exhibit the variation of the Frolicher spectral sequence along curves of complex structures on a real 6-dimensional manifold.

Lattices and Periodic Geodesics in Pseudoriemannian 2-step Nilpotent Lie Groups

We give a basic treatment of lattices Γ in these groups. Certain tori TF and TB provide the model fiber and the base for a submersion of Γ\N. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.

SYMMETRIES OF SECTIONAL CURVATURE ON 3-MANIFOLDS

In dimension three, there are only two signatures of metric tensors: Lorentzian and Riemannian. We find the possible pointwise symmetry groups of Lorentzian sectional curvatures considered as rational functions, and determine which can be realized on naturally reductive homogeneous spaces. We also give some examples. MSC (1991): Primary 53C50; Secondary 53B30, 53C30. 1Supported by Project XUGA8050189, Xunta de Galicia, Spain. 2On leave from Math. Dept., Wichita State Univ., Wichita KS 67260, U.S.A., pparker@twsuvm.uc.twsu.edu 3Partially supported by DGICYT-Spain.

Properties of the basic cohomology of transversely Kähler foliations

In this paper we study the complex basic cohomology of transversely Hermitian foliations. We use the methods developed in [7] and prove that for transversely Kähler foliations the foliated version of the Frölicher spectral sequence collapses at the first level and that the minimal model for the complex basic cohomology is formal. To stress that these properties are particular to transversely Kähler foliations we construct examples of transversely Hermitian foliations for which these theorems do not hold.

Dolbeault homotopy theory and compact nilmanifolds

In this paper we study the degeneration of both the cohomology and the cohomotopy Frolicher spectral sequences in a special class of complex manifolds, namely the class of compact nilmanifolds endowed with a nilpotent complex structure. Whereas the cohomotopy spectral sequence is always degenerate for such a manifold, there exist many nilpotent complex structures on compact nilmanifolds for which the classical Frolicher spectral sequence does not collapse even at the second term.