On neighborhoods of embedded complex tori

TL;DR

This paper proves that under certain conditions, an embedded complex torus in a complex manifold has a neighborhood biholomorphic to a neighborhood in its normal bundle, extending classical results to higher dimensions with specific geometric constraints.

Contribution

It establishes a biholomorphic equivalence between neighborhoods of embedded complex tori and their normal bundles under non-resonant Diophantine conditions and split tangent bundles.

Findings

Neighborhood biholomorphism is achieved under Diophantine conditions.

The result generalizes classical theorems to higher-dimensional complex tori.

Conditions on the normal bundle's transition functions are crucial.

Abstract

The goal of the article is to show that an n-dimensional complex torus embedded in a complex manifold of dimensional n+d, with a split tangent bundle, has neighborhood biholomorphic a neighborhood of the zero section in its normal bundle, provided the latter has (locally constant) Hermitian transition functions and satisfies a non-resonant Diophantine condition.