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 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, 0) → (C, 0), f is real, we define a complex analytic family of quadratic forms parameterized by the points of the image (C, 0) of the map f , which become real for real and in this case their signatures defer from the “real” index by χ(V )− 1, where χ(V ) is the Euler characteristic of the corresponding smoothing V = f −1( ) ∩Bδ of the icis V . 2000 Math. Subj. Class. 14B05, 32S99.