On Gauss-Bonnet and Poincar\'e-Hopf type theorems for complex $\partial$-manifolds

TL;DR

This paper establishes Gauss-Bonnet and Poincaré-Hopf theorems for complex partial manifolds derived from compact manifolds by removing divisors with isolated singularities, extending classical results to singular settings.

Contribution

It introduces new theorems for complex partial manifolds with divisors having isolated singularities, generalizing classical geometric theorems to singular complex varieties.

Findings

Proves Gauss-Bonnet theorem for complex partial manifolds with isolated singularities.

Establishes Poincaré-Hopf theorem in the context of singular divisors.

Extends classical theorems to cases with divisors decomposed into parts with isolated singularities.

Abstract

We prove a Gauss-Bonnet and Poincar\'e-Hopf type theorems for complex $\partial$-manifold $\tilde{X} = X - D$, where $X$ is a complex compact manifold and $D$ is a reduced divisor. We will consider the cases such that $D$ has isolated singularities and also if $D$ has a (not necessarily irreducible) decomposition $D=D_1\cup D_2$ such that $D_1$, $D_2$ have isolated singularities and $C=D_1\cap D_2$ is a codimension $2$ variety with isolated singularities.