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"Complex manifolds, foliations by complex leaves and their deformations" (DEFFOL)
Start date: Sep 1, 2011, End date: Aug 31, 2013 PROJECT  FINISHED 

"My proposal is to investigate some new aspects of the geometry of complex manifolds, foliations by complex leaves and their deformations. Emphasis is put on the relations between these three themes.The basic objects of complex geometry are compact complex manifolds. However, only very special classes are well understood. And the only theory which applies to the general case is Kodaira-Spencer theory of deformations. To each structure of a compact complex manifold it associates a finite-dimensional space, called Kuranishi space, containing all the small deformations of this initial structure. Although it is a very classical theory, almost nothing is known about the geometry of this space. Last year, I discovered that it has a natural foliated structure and I want to study it more thoroughly.On the other hand, Kuranishi spaces exist for transversely holomorphic foliations, which form a generalization of complex structures. There is however another natural generalization, that of foliation by complex leaves. Now in this case there is no Kuranishi space (the finite-dimensionality fails). I have some precise ideas to construct (finite-dimensional) Kuranishi spaces in a generalized sense.Finally I also want to introduce a notion of geometric rigidity, which is better adapted to the case of a foliation than the classical notion of rigidity.As a researcher, I work for twelve years on the geometry and topology of compact complex manifolds. But to achieve these goals, I need to strengthen my training in deformation theory. So I plan to spend two years at CRM, Barcelona, under the guidance of Marcel Nicolau, an expert in deformations of complex manifolds and transversely holomorphic foliations. I could also benefit from the intense mathematical activity of the CRM, especially in complex analysis and geometry.Achieving these goals would open new lines of research and qualify me as a leading expert in the field. Last but not least, I could look for another position."
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