Defesa de Tese.

Banca examinadora: Hossein Movasati - Orientador - IMPA Alcides Lins - IMPA Roberto Villaflor - IMPA Jin Cao - Universidade Tsinghua, China Stefan Reiter (Universidade Bayreuth, Alemanha) Younes Nikdelan - Suplente - UERJ

Abstract: This thesis is devoted to the study of Hodge cycles of perturbations of a Fermat variety and its periods. Periods (roughly speaking multiple integrals) are an essential part of Hodge theory. Periods of algebraic cycles encode algebraic dependence relations between transcendental numbers (usually coming from values of special functions). This would be also true for Hodge cycles if the Hodge conjecture holds true. We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by simple. We explore some examples and provide a method to calculate explicitly a set of generators of this subspace. As an application, with these explicit Hodge cycles, we find expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable (the special functions of the previous paragraph in this context) .