Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle

TL;DR

This paper generalizes Ueda's theory to higher codimensions, providing conditions for holomorphic foliations near a compact submanifold with flat normal bundle, with applications to Hermitian metrics on line bundles.

Contribution

It extends Ueda's linearizability results to higher codimensional cases and links these to the existence of semi-positive Hermitian metrics on nef line bundles.

Findings

Established a sufficient condition for holomorphic foliation existence near the submanifold.

Connected the foliation theory to the positivity of line bundles.

Provided new insights into the neighborhood structure of submanifolds with flat normal bundles.

Abstract

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher-codimensional generalization of Ueda's result, we give a sufficient condition for the existence of a non-singular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semi-positive curvature on a nef line bundle.