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

2018/08/30 by Koike, Takayuki, Ogawa, Noboru
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.10219

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 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 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 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.

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