On Gauss-Bonnet and Poincaré-Hopf type theorems for complex ∂-manifolds

Abstract

We prove a Gauss-Bonnet and Poincar´e-Hopf type theorem for complex ∂-manifold 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 = D1 ∪ D2 such that D1, D2 have isolated singularities and C = D1 ∩ D2 is a codimension 2 variety with isolated singularities.