Gauss Bonnet Theorem for Singular Curves in CPn

Abstract

This paper provides a generalization of the Gauss-Bonnet theorem to singular irreducible projective algebraic curves. For any singular curve C with singularity set Sigma, we prove that if one integrates the Gaussian curvature provided by restricting the Fubini-Study metric to the noncompact manifold CsetminusSigma that the result is well defined and equal 2pi times the Euler characteristic of C's desingularization plus finitely many correction terms specified by the structure of C's singularities.