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Poisson geometry and Lefschetz fibrations

Grant number: 22/04705-8
Support Opportunities:Scholarships in Brazil - Post-Doctorate
Effective date (Start): September 01, 2022
Effective date (End): August 31, 2024
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Cristián Andrés Ortiz González
Grantee:Daniel Felipe López Garcia
Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In this postdoctoral project we study Lefschetz fibrations over a special class of symplectic manifolds given by symplectic leaves of Poisson Lie groups. Our main goal consists on the construction of new examples of Lefschetz fibrations over symplectic manifolds, relying on techniques which combine Lie Theory and Poisson Geometry. On the other hand, as a continuation of previous works, we have other issues that will be addressed. Firstly, we want to use the Picard-Lefscherz Therory, as in the case of the mirror quintic Calabi-Yau threefold, to study homology cycles supported on Lagrangian submanifolds in hypersurfaces of the n-projective spaces, for example the Fermat variety. Secondly, in a previos article, it was proved that a Poisson manifold with a Hausdorff integration admits a Hausdorff complete symplectic realization. An interesting question which we would like to work on is the converse of this Corollary, more precisely: does a Hausdorff symplectic complete realization give rise to a Hausdorff symplectic groupoid? Finally, in the context of holomorphic foliations, we study the problem of centers in the projective space of dimension 2. More precisely, we investigate the irreducible component formed by the logarithmic foliations but in the projective case. (AU)

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