2001/05/29 by Ebeling, Wolfgang, Gusein-Zade, Sabir M. · 1 citation
#14B05 #32S99 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0105242
There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on Rn 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:(Cn, 0) → (Ck, 0), f is real, we define a complex analytic family of quadratic forms parameterized by the points ε of the image (Ck, 0) of the map f, which become real for real ε and in this case their signatures defer from the "real" index by χ(Vε)-1, where χ(Vε) is the Euler characteristic of the corresponding smoothing Vε=f-1(ε)∩ Bδ of the icis V.