Arnolds' seminar
October 9, 2001.
S.M.Gusein-Zade
Indices of 1-forms on an isolated complete intersection singularity

There are some generalizations of the classical
Eisenbud--Levine--Khimshiashvili formula for the index of a singular point
of an analytic vector field on $R^n$ to 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 {\bf icis} $V$.
The talk is the result of a joint work with W.Ebeling.