7
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.