For an isolated hypersurface singularity, generic holomorphic vector fields with isolated zeros have GSV-index at least 1 + (-1)^dim(V) * τ(V,0), with equality exactly when the field extends to a nondegenerate zero in ambient space.
and Saravia-Molina, N.: On some indices of foliations and applications
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Generic vector fields on isolated complex hypersurface germs
For an isolated hypersurface singularity, generic holomorphic vector fields with isolated zeros have GSV-index at least 1 + (-1)^dim(V) * τ(V,0), with equality exactly when the field extends to a nondegenerate zero in ambient space.