Indices of 1-forms on an isolated complete intersection singularity

Abstract

There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an icis $V=f^{-1}(0)$, $f:(C^n, 0) \to (C^k, 0)$, $f$ is real, we define a complex analytic family of quadratic forms parameterized by the points $\epsilon$ of the image $(C^k, 0)$ of the map $f$, which become real for real $\epsilon$ and in this case their signatures defer from the "real" index by $\chi(V_\epsilon)-1$, where $\chi(V_\epsilon)$ is the Euler characteristic of the corresponding smoothing $V_\epsilon=f^{-1}(\epsilon)\cap B_\delta$ of the icis $V$.