Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations

Abstract

AbstractLet C be a smooth elliptic curve embedded in a smooth complex surface X such that C is a leaf of a suitable holomorphic foliation of X. We investigate the complex analytic properties of a neighborhood of C under some assumptions on the complex dynamical properties of the holonomy function. As an application, we give an example of {(C,X)} in which the line bundle {[C]} is formally flat along C, however it does not admit a {C^{\infty}} Hermitian metric with semi-positive curvature. We also exhibit a family of embeddings of a fixed elliptic curve for which the positivity of normal bundles does not behave in a simple way.