The index of a real vector field at an isolated complete intersection singularity

TL;DR

This paper develops a formula for the index of an analytic vector field at an isolated zero on a real analytic hypersurface with a singularity, extending residue-based methods to the complete intersection case.

Contribution

It introduces a generalized residue symbol approach for calculating the index of vector fields on singular complete intersections, expanding previous smooth case formulas.

Findings

Derived a residue-based formula for the index at singularities.

Extended bilinear form calculus to complete intersection singularities.

Connected the index calculation to local residue symbols and free resolutions.

Abstract

In an unpublished note [H1] we have described a method to obtain a formula for the index of an analytic vector field with (complex) isolated zero on a real analytic hypersurface with (complex) isolated singularity. This formula, like the one of Eisenbud-Levine and Khimshiashvili [AGV] for smooth points, expresses the index by the signature of bilinear forms, which are defined by a local residue symbol (cf. [Ma]). In the complete intersection case, we use a generalized residue symbol, defined for free resolutions in [LJ], in the special case of generalized Koszul complexes to obtain a suitable calculus for the bilinear forms involved.