On the neighborhood of a torus leaf and dynamics of holomorphic foliations

Takayuki Koike; Noboru Ogawa

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arXiv · 1808.10219

On the neighborhood of a torus leaf and dynamics of holomorphic foliations

Takayuki Koike·Noboru Ogawa

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Abstract

Let $X$ be a complex surface and $Y$ be an elliptic curve embedded in $X$. Assume that there exists a non-singular holomorphic foliation $\mathcal{F}$ with $Y$ as a compact leaf, defined on a neighborhood of $Y$ in $X$. We investigate the relation between Ueda's classification of the complex analytic structure of a neighborhood of $Y$ and complex dynamics of the holonomy of $\mathcal{F}$ along $Y$. More precisely, we show that the pair $(Y,X)$ is of type ($\gamma$) in his classification when there exists a closed curve in $Y$ along which the holonomy of $\mathcal{F}$ is irrationally indifferent and non-linearizable. We also investigate the metric semi-positivity of the line bundle determined by the divisor $Y$. Our approach is based on the theory of hedgehogs, due to P\'{e}rez-Marco.

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Takayuki Koike, Noboru Ogawa. 2018-08-30. On the neighborhood of a torus leaf and dynamics of holomorphic foliations. https://arxiv.org/abs/1808.10219

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