  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.
