Defesa de Tese.

Banca examinadora: Hossein Movasati - Orientador - IMPA Alcides Lins - IMPA Younes Nikdelan - UERJ Colin Christopher - Univ. Phymouth Lubomir Gavrilov - Université Toulouse Roberto Villaflor - Suplente - IMPA

Abstract: The monodromy action relates the topology of a parametrized family of manifolds with the fundamental group of the parameters space. There is a topological approach based on the Lefschetz fibrations. Given a smooth fibration, the action of the fundamental group of the base without the critical values on the homology group of a regular fiber is the monodromy action. In some cases, the homology group is generated by the so-called vanishing cycles; which are associated with the critical values of the fibration. Another method is by using Picard-Fuchs equations, whose solutions are the periods of some complex manifolds, and the monodromy action is obtained by holomorphic continuation around to the critical values. The first main part is related to symplectic geometry by studying the Lagrangian cycles in a family of mirror quintic Calabi-Yau threefolds. Considering a symplectic structure in a fibration such that the fibers are symplectic manifolds, it is possible to show that the vanishing cycles are Lagrangian submanifolds and the monodromy action is given by symplectomorphisms. In the case of mirror quintic, there is an explicit form for the monodromy matrices and for two Lagrangian cycles which are supported in a 3-sphere and 3-torus. We study the orbit of these cycles by monodromy action. The second main part is related to the monodromy problem in polynomial foliations. Given a polynomial with two variables we consider its associated foliation. The center points of the foliation are vanishing cycles. The monodromy problem is to establish conditions on the polynomial so the orbit of the vanishing cycles generate the whole homology group. It is possible to relate the monodromy action with a diagram called the Dynkin diagram. Thus, given a polynomial, we translate questions on the subspaces generated by these orbits to combinatory aspects of the diagram .