A Hilbert bundle approach to the sharp strong openness theorem and the Ohsawa-Takegoshi extension theorem

TL;DR

This paper introduces a Hilbert bundle framework to advance complex analysis, leading to generalized theorems including a sharp strong openness theorem and an improved Ohsawa-Takegoshi extension theorem, with additional alternative proofs.

Contribution

It develops a novel Hilbert bundle approach to complex Brunn-Minkowski theory, resulting in generalized and sharper versions of key theorems in complex analysis.

Findings

Proved a generalized sharp strong openness theorem.

Established a sharp Ohsawa-Takegoshi extension theorem.

Provided a second proof of Guan-Zhou's strong openness theorem.

Abstract

The following paper is around parts of the first named author's thesis. We discuss (what we call) a Hilbert bundle approach to complex Brunn-Minkowski theory and obtain a general monotonicity theorem. As two applications, we prove a generalization of Guan's sharp strong openness theorem and a sharp Ohsawa-Takegoshi extension theorem. A second proof of Guan-Zhou's strong openness theorem using a Donnelly-Fefferman estimate is also given.