# holomorphic vector fields tangent to a hypersurface

Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $\{f=0\}$, $f$ holomorphic. A (germ of) holomorphic vector field $X$ leaves $V$ invariant if ${\rm d}f(X)$ belongs to the ideal generated by $f$.

My question is: For any such $V$, does there exist a vector field tangent to $V$ with an isolated singularity?

When $V$ is smooth or $V$ has an isolated singularity (at the origin) then one can readily exhibit such vector fields. However, for germs of non-isolated singularities I could neither prove this nor find a counterexample.

• What is the domain and codomain of $f$? If $f: \mathbb{C}^3 \to \mathbb{C}$, isolated singularities are all removable. – AmorFati Nov 19 '17 at 11:24
• @KyleBroder, $f$ is a germ of holomorphic function $f\colon (\mathbb{C}^n ,0) \rightarrow (\mathbb{C},0)$. By singularity here, I mean the singularities of $V= f^{-1}(0)$ i.e. the critical points of $f$. – Alan Muniz Nov 20 '17 at 15:04