Holomorphic vector fields tangent to a hypersuface singularity

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.

1 Answer

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.

• Thank you for the answer. That is a nice application of the cross ratio. – Alan Muniz Oct 6 '18 at 1:09