An Ohsawa-Takegoshi theorem on compact K\"ahler manifolds

TL;DR

This paper proves a version of the Ohsawa-Takegoshi extension theorem on compact K"ahler manifolds, addressing technical challenges in regularization and providing a partial result with natural additional hypotheses.

Contribution

It extends the Ohsawa-Takegoshi theorem to compact K"ahler manifolds under new conditions, overcoming regularization complications.

Findings

Established a partial Ohsawa-Takegoshi theorem on compact K"ahler manifolds.

Identified natural additional hypotheses necessary for the proof.

Highlighted the technical challenges in regularizing quasi-psh functions.

Abstract

In this article we prove a theorem of Ohsawa-Takegoshi type on compact K\"ahler manifolds. Our arguments follow the "standard" approach for this kind of extension results; however, there are many complications arising from the regularization process of quasi-psh functions on compact K\"ahler manifolds, and unfortunately we only obtain a particular case of the expected result. We remark that the additional hypothesis we are forced to make are natural, since they are in many situations; we hope to remove them in a near future.