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 $V$.

19 pages

Cited by in corpus (5)

• Determinantal singularities and Newton polyhedra
• Morse numbers of function germs with isolated singularities
• Radial index and Poincaré-Hopf index of 1-forms on semi-analytic sets
• Homological index for 1-forms and a Milnor number for isolated singularities
• Homological indices of collections of 1-forms