Edit: This is a real coefficient version of the current post.

Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow?

There is a regular leaf $L$ whose holonomy, along at least one closed curve on it, is not trivial but $L$ does not intersect the real part $im (z)=im(w)=0,\;(z,w) \in \mathbb{C}^2$.


A leaf with non trivial holonomy is called a complex limit cycle, according to the terminology used in the video lecture by Ilyashenko described in the following answer:

The error in Petrovski and Landis' proof of the 16th Hilbert problem

  • 2
    $\begingroup$ When I saw your question I did the bet that you would eventually, at some point, edit its tags just to add "limitcycle". It's too predictable :) $\endgroup$
    – YCor
    Feb 18 '18 at 0:30
  • $\begingroup$ @YCor Yes. As you have predicted, I could not resist myself from adding this tag:) $\endgroup$ Feb 18 '18 at 5:32
  • 3
    $\begingroup$ @YCor I will just add that (limitcycle) seems to be renamed to (limit-cycles). I suppose that was your suggestion on meta - although I no longer can see the post, since it was deleted. $\endgroup$ Jan 20 '19 at 11:29

A revision: Novembre 2020

I am realy indebted to Loic Teyssier for his $2$ very valuable comments and suggestions. I summarize his comments as follows:

  1. To have a hyperbolic complex limit cycle it is not sufficient you check $\int_{\gamma} \alpha \neq 0$ but you should also check that this integral is different from $2k\pi i$

  2. If your example realy work, you can obtain a similar example with real coefficient if you replace $z^2+w^2+1=0$ with $(z^2+w^2-4i=0)$.

Now after more than one year of his suggestions, I look at my answer again.

His first comment leads me to compute the corresponding integral again. Then I just realiz that this integral is equal to $0$!. More over I realize that not only this example is not appropriate for the purpose of this question but also every possible reform of this example is not appropriate. For example consideration of $$ \begin{cases} z'=w+z(z^2+w^2-4i)\\ w'=-z+w(z^2+w^2-4i) \end{cases}$$ does not work. For all these examples the holonomy would be tangent to the identity maps. Hence a relevant question would be that: Are the corresponding leaf $z^2+w^2=4i$ is a leaf with non trivial holonomy?

His second comment help me to realize that the following system has a complex limit cycle $z^2+w^2+1=0$ which obviously does not intersect the real plane $\mathbb{R}^2$. Here is the true example as required as an answer to this post:

$$ \begin{cases} z'=w+z(z^2+w^2+1)\\ w'=-z+w(z^2+w^2+1) \end{cases}$$

Finally we include the following question in our answer:

Can a real polynomial vector field possess a hyperbolic complex limit cycle $\gamma$ which is not algebraic and does not intersect the real plane?

The previous version of my answer:

The answer to this question is yes. There is a complex polynomial vector field on $\mathbb{C}^2$ with a complex limit cycle which does not intersect the real plane $im(z)=im(w)=0$.

Consider the differential equation $$\begin{cases}z'=w+(z^2+w^2-4i)\\ w'=-z+(z^2+w^2-4i) \end{cases}$$

The regular leaf $L: z^2+w^2=4i$ of this singular foliation does not intersect the real part of $\mathbb{C}^2$. This leaf, which is topologically a cylinder, has a non trivial holonomy. In fact we have more: there is a closed curve on this leaf whose corresponding holonomy map is a hyperbolic map: namely the holonomy is not tangent to the identity map. Here is the argument:

The hyperbolicity, hence non triviality, of the holonomy of this leaf is a consequence of Theorem 3.2 Page 333 of the paper: First Variation of Holomorphic forms and some applications.

Elaboration: The foliation is defined by $$\omega= (w+(z^2+w^2-4i))dw-(-z+(z^2+w^2-4i))dz=0$$

To apply the theorem 3.2 in the above paper we find a $1-$ form $\alpha$ which satisfies $d\omega=\alpha \wedge \omega$, locally around an appropriate closed curve $\gamma$ in $L$.

Represent the above $1$- form $\omega$ in the form $\omega=Pdw-Qdz$. Then for $$\alpha=(P_z+Q_w)/(P^2+Q^2)(Pdz+Qdw)$$ we have $d\omega=\alpha \wedge \omega$. Note that $P^2+Q^2$ does not vanish on $L$. Now we have to compute $\int_{\gamma} \alpha$, along an appropriate closed curve $\gamma \subset L$, and show that this integral is non zero.

To compute this integral we parametrize the cylinder $L$ with
$$ \phi(t)= \begin{cases} z(t)=t+i/t\\w(t)=t/i+1/t \end{cases}$$ where $$\phi:\mathbb{C}\setminus \{0\}\to \mathbb{C}^2$$ is the global parametrization of $L$. We will see that the desired appropriate curve $\gamma$ is $\phi(S^1)$.

We denote by $\phi^*(\alpha) $, the pull back of $\alpha$ under embedding $\phi$. Now a very simple computation shows that $\int_{S^1} \phi^* \alpha$ is non zero since we obtain a pole of order 1 at the origin. In fact the later integral is $\int_{S^1} 2(z(t)+w(t))(wdz-zdw)$. An straightforward and short computation shows that we have a non degenerate pole, namely a pole of order 1. so the integral does not vanish. So the multiplier $e^{\int _{S^1} \alpha} $ is different from $1$. Then the leaf $L$ is a hyperbolic leaf. $\square$


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