A Characterization of regular points by $L^2$ Extension Theorem

TL;DR

This paper establishes a characterization of regular points in complex analytic sets using the $L^2$ extension theorem, linking geometric regularity to analytic extension properties.

Contribution

It provides a new criterion for regularity of complex analytic set germs based on the validity of the $L^2$ extension theorem.

Findings

Germ of a complex analytic set is regular iff the $L^2$ extension theorem holds.

Derived a necessary condition for extending bounded holomorphic sections from singular sets.

Abstract

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related $L^2$ extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.