On Gauss–Bonnet and Poincaré–Hopf type theorems for complex $\partial$-manifolds

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.