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.