Poisson geometry and Lefschetz fibrations

• Grant number: 22/04705-8
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2022
• End date: August 31, 2025
• 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)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

LOPEZ-GARCIA, DANIEL; VALENCIA, FABRICIO. ON THE MONODROMY ACTION FOR f(x, y) = g(x) plus h(y). Proceedings of the American Mathematical Society, v. 153, n. 5, p. 12-pg., 2025-03-05. (20/07704-7, 22/04705-8)

GARCIA, DANIEL LOPEZ. MONODROMY PROBLEM AND TANGENTIAL CENTER-FOCUS PROBLEM FOR PRODUCTS OF LINES IN GENERAL POSITION IN P². MOSCOW MATHEMATICAL JOURNAL, v. 24, n. 2, p. 19-pg., 2024-04-01. (22/04705-8)

BRAUN, ANDREAS P.; FORTIN, HUGO; GARCIA, DANIEL LOPEZ; LOYOLA, ROBERTO VILLAFLOR. More on G-flux and general hodge cycles on the Fermat sextic. Journal of High Energy Physics, v. N/A, n. 6, p. 47-pg., 2024-06-07. (22/04705-8)