Residue formulas for logarithmic foliations and applications

Abstract

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