# Moscow Mathematical Journal

Volume 21, Issue 3, July–September 2021 pp. 493–506.

On Gauss–Bonnet and Poincaré–Hopf Type Theorems for Complex $\partial$-Manifolds

**Authors**:
Maurício Corrêa (1), Fernando Lourenço (2), Diogo Machado (3), and Antonio M. Ferreira (4)

**Author institution:**(1) Icex – UFMG, Av. Antônio Carlos 6627, 30123-970, Belo Horizonte-MG, Brazil

(2) DEX – UFLA, Campus Universitário, Lavras MG, Brazil, CEP 37200-000

(3) DMA – UFV, Avenida Peter Henry Rolfs, s/n – Campus Universitário, 36570-900 Vi cosa-MG, Brazil

(4) DEX – UFLA, Campus Universitário, Lavras MG, Brazil, CEP 37200-000

**Summary: **

We prove a Gauss–Bonnet and Poincaré–Hopf type theorem for complex $\partial$-manifold $\widetilde{X} = X - D$, 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.

2020 Math. Subj. Class. Primary: 32S65, 32S25, 14C17

**Keywords:**Logarithmic foliations, Gauss–Bonnet type theorem, Poincaré–Hopf index, residues.

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