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 $\mathbb R^n$ 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 $V=f^{-1}(0)$, $f: (\mathbb C^n,0)\to(\mathbb C^k,0)$, $f$ is real, we define a complex analytic family of quadratic forms parameterized by the points $\varepsilon$ of the image $(\mathbb C^k,0)$ of the map $f$ which become real for real $\varepsilon$ and in this case their signatures defer from the “real” index by ${}_\mathcal X (V_\varepsilon)-1$, where ${}_\mathcal X(V_\varepsilon)$ is the Euler characteristic of the corresponding smoothing $V_\varepsilon= f^{-1}(\varepsilon)\cap B_\delta$ of the icis $V$.