An important diffuculty in complex geometry is that there is only a few known examples of compact complex manifolds which are non-algebraic or even non-Kählerian. In recent papers J.J Loeb - M- Nicolau and S. López de Medrano - A Verjovsky developed a new method of construcction of such manifolds which relies in the dynamical properties of certain holomorphic vector fields. Our aim in this project is to understand better this interplay between geometry and dynamics and to extend the above construction in several directions. The manifolds so constructed are non-kählerian but are naturally endowed with a non-vanishing holomorphic vector field which is transversely algebraic in a precise sense. We want to proof that in fact they are characterised by this property. This should provide a new insight into the difficult question of the classification of compact complex manifolds of low dimension. The other objectives of the project concem the deformations of geometric structures, principally of the singularities of holomorphic vector fields in dimension two. We propose, in particular, the study of the unfoldings of these singularities by means of: (I) methods of CR geometry and contact geometry and contact geometry applied to the CR line bundle induced by the vector field in a small sphere centered at the singularity and (II) the construction of an analogue of the Teichmüller space based on the properties of quasi-conformal mappings and a precise analysis of the irreductible singularities. In both cases the aim is to prove the existence of a finite dimensional versal unfolding (ot moduli space) for singular foliations.
|Effective start/end date||19/06/01 → 19/06/04|
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