Generic vector fields on isolated complex hypersurface germs

We study holomorphic vector fields on isolated hypersurface singularities and derive global obstructions to the existence of holomorphic vector fields on compact singular varieties. For a hypersurface germ $(V,0)$ with an isolated singularity, we characterize the generic elements in the space of holomorphic vector fields with isolated singularity in terms of the GSV-index. Letting $\tau(V,0)$ denote the Tjurina-Greuel number, we prove that the minimal possible index is bounded below by $1+(-1)^{\dim(V)}\tau(V,0)$. We further prove that equality holds if the vector field admits an extension to $\mathbb{C}^{n+1}$ with a nondegenerate singularity at $\underline{0}$ and, in the case $n$ is odd, that such extensions, when they exist, form an open dense subset of the set of vector fields with an isolated singularity at $\underline{0}$. This yields a description of the generic vector fields on weighted homogeneous hypersurface germs. As a consequence, we obtain a characterization of weighted homogeneous hypersurface germs. Also, as applications to singular hypersurfaces in complex manifolds, we derive constraints on compact singular varieties admitting holomorphic vector fields. In particular, we show that an irreducible compact singular complex curve carrying a nontrivial holomorphic vector field with zeros is rational and has at most two singular points. We further prove that, for singular surfaces in K\"{a}hler 3-folds satisfying suitable positivity assumptions on the adjoint line bundle, the geometric genus is greater of equal than the irregularity.