Sobre a conjectura da monodromia

Abstract

Consider a holomorphic map that leaves the complex Cartesian n-product for the complex set, we will try to see relation between the monodromy action that is the automorphism in the group of homology of the Milnor fi ber induced by a generator of the fundamental group of the complex set minus the origin and Igusa’s zeta function obtained through the embedding resolution. More specifi cally, we shall look at the monodromy conjecture which says that whenever a point s 0 is pole of Igusa’s zeta function, then the exponential of 2i pis 0 is the eigenvalue of the monodromy action. We will use the fact that the case n = 2 has already been fully demonstrated to show an idea of the proof for some cases of n = 3, such as homogeneous polynomials and super-isolated singularities.