Residue Formulas for logarithmic Foliations and applications

Abstract

In this work we prove a Baum-Bott type formula for non-compact complex manifold of the form $\tilde{X}=X-\mathcal{D}$, where $X$ is a complex compact manifold and $\mathcal{D}$ is a normal crossing divisor on $X$. As applications, we provide a Poincaré-Hopf type Theorem and an optimal description for a smooth hypersurface $\mathcal{D}$ invariant by an one-dimensional foliation $\mathcal{F}$ on ${\mathbb{P}}^n$ satisfying $Sing(\mathcal{F})⊊\mathcal{D}.$

20 pages, to appear in Transactions of the American Mathematical Society. Major changes following suggestions of the referees