Holomorphic vector fields tangent to a hypersuface singularity

Question

Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $V =\{f=0\}$, $f$ a germ of holomorphic function. A (germ of) holomorphic vector field $X$ on $(\mathbb{C}^n,0)$ is "tangent to $V$" if $X(f) = {\rm d}f(X)$ belongs to the ideal generated by $f$ in $\mathcal{O}_{\mathbb{C}^n,0}$.

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

If $V$ is smooth or has an isolated singularity, this is fairly simple. However, for germs of non-isolated hypersurface singularities I could neither prove this nor find a counterexample.

Answer 1

I think that the answer to this question is no for $\mathbb C^3$, for $$f=zy(z-y)(z-xy).$$ I'll assume that $v$ is holomorphic and tangent to $f=0$ in a small neighbourhood of $(0,0,0)$.

Proof. Suppose by contradiction that $v$ is a holomorphic vector field tangent to $f=0$ near $(0,0,0)$ with an isolated zero at $(0,0,0)$. Note that the surface $f=0$ has a singularity along the $x$-axes. So $v$ should be tangent to the $x$-axes and also should be non-zero on it by our assumptions. I claim that this is not possible.

Indeed, note that the tangent cone of the surface $f=0$ at a point $(x_0,0,0)$ is the union of $4$ planes, $z=0$, $y=0$, $z-y=0$ and $z-x_0y=0$. These planes have double ratio $x_0$ (to define such a double ratio intersect these $4$ planes with any plane passing through $(x_0,0,0)$ and take the double ratio of $4$ lines in the intersection). On the other hand if $v$ were non-zero on the $x$-axis, its flow would send points of the $x$-axes to different points and so it would not preserve the double ratio. This is a contradiction, since a byholomorphism must preserve such a ratio. QED.