On a neighborhood of a torus leaf of a certain class of holomorphic foliations on complex surfaces

TL;DR

This paper studies the complex analytic structure of neighborhoods around elliptic curve leaves in holomorphic foliations on complex surfaces, revealing nuanced properties of line bundles and holonomy under certain dynamical conditions.

Contribution

It provides new insights into the local complex geometry near elliptic leaves and constructs examples illustrating subtle differences between formal flatness and smooth positivity of line bundles.

Findings

Neighborhoods exhibit specific complex analytic properties influenced by holonomy dynamics.

Existence of line bundles that are formally flat but lack semi-positive Hermitian metrics.

Examples demonstrate the divergence between formal flatness and smooth metric positivity.

Abstract

Let $C$ 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 complex analytic properties of a neighborhood of $C$ under some assumptions on 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.