Residue Formulas for logarithmic Foliations and applications

TL;DR

This paper develops residue formulas for logarithmic foliations on non-compact complex manifolds, deriving a Baum-Bott type formula and applications like a Poincaré-Hopf theorem and descriptions of invariant hypersurfaces.

Contribution

It introduces a Baum-Bott type residue formula for non-compact complex manifolds with divisors, extending classical results to new geometric settings.

Findings

Proves a Baum-Bott type formula for non-compact manifolds with divisors.

Provides a Poincaré-Hopf type theorem for logarithmic foliations.

Characterizes invariant hypersurfaces under certain foliations.

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 Poincar\'e-Hopf type Theorem and an optimal description for a smooth hypersurface $\mathcal{D}$ invariant by an one-dimensional foliation $\mathscr F$ on $\mathbb{P}^n$ satisfying $Sing(\mathscr F) \subsetneq \mathcal{D}.$