Resíduos e classes características para folheações do tipo logarítmicas

Abstract

In the first part of this thesis we consider the problem of finding versions of Baum-Bott index theorem for non-compact complex manifolds of type ~X = X n D, where X is a compact complex manifold and D is a divisor on X. We show such versions in the case where D has singularities normal crossing type or when D has isolated singularities. This allows us to determine when a smooth hypersurface, invariant under a one-dimensional foliation F in Pn, contains or not all the singularities of F. Moreover, we can recover Soares quota for the Poincaré problem in this context. In the second part, we define the GSV index for Pfaff systems whose invariant variety has codimension equal to its rank. Finally, we show that the non-negativity of this index gives us the obstruction to the Poincaré problem solution for Pfaff systems.