On the neighborhood of a torus leaf and dynamics of holomorphic foliations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916826909245440 |
|---|---|
| author | Koike, Takayuki Ogawa, Noboru |
| author_facet | Koike, Takayuki Ogawa, Noboru |
| contents | 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 ($γ$) 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 Pérez-Marco. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_10219 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the neighborhood of a torus leaf and dynamics of holomorphic foliations Koike, Takayuki Ogawa, Noboru Complex Variables Dynamical Systems 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 ($γ$) 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 Pérez-Marco. |
| title | On the neighborhood of a torus leaf and dynamics of holomorphic foliations |
| topic | Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/1808.10219 |