Euler obstruction and Lipschitz-Killing curvatures

TL;DR

This paper links the Euler obstruction of complex analytic germs to Lipschitz-Killing curvatures and Chern forms, providing new characterizations and answering a question of Fu on the Euler obstruction and Gauss-Bonnet measure.

Contribution

It introduces novel characterizations of the Euler obstruction using Lipschitz-Killing curvatures and extends results to the global case, addressing Fu's question.

Findings

Euler obstruction characterized by Lipschitz-Killing curvatures and Chern forms

Positive answer to Fu's question on Euler obstruction and Gauss-Bonnet measure

Analogous results established for global Euler obstruction

Abstract

Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstruction. As a corollary, we give a positive answer to a question of Fu on the Euler obstruction and the Gauss-Bonnet measure.