Tohoku Mathematical Journal
2025
September
SECOND SERIES VOL. 77, NO. 3

Tohoku Math. J.
77 (2025), 421-443

Title ON THE NEIGHBORHOOD OF A TORUS LEAF AND DYNAMICS OF HOLOMORPHIC FOLIATIONS

Author Takayuki Koike and Noboru Ogawa

(Received May 16, 2023, revised November 8, 2023)
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 Pérez-Marco.

Mathematics Subject Classification. Primary 37F75; Secondary 37F50.
Key words and phrases. Ueda's neighborhood theory, dynamics of holomorphic foliations, hedgehogs.

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