On $L^2$ extension from singular hypersurfaces

TL;DR

This paper links the singularities of measures used in $L^2$ extension theorems to algebraic geometry concepts, providing new insights and proofs, and clarifying limitations of extending functions from singular hypersurfaces.

Contribution

It identifies the singularity of the Ohsawa measure with algebraic pair singularities, offers an analytic proof of inversion of adjunction, and compares various $L^2$ extension results, including limitations.

Findings

Singularity of Ohsawa measure corresponds to algebraic pair singularities.

Provides an analytic proof of inversion of adjunction.

Generalizes a negative result limiting $L^2$ extension improvements.

Abstract

In $L^2$ extension theorems from a singular hypersurface in a complex manifold, important roles are played by certain measures such as the Ohsawa measure which determine when a given function can be extended. We show that the singularity of the Ohsawa measure can be identified in terms of singularity of pairs from algebraic geometry. Using this, we give an analytic proof of the inversion of adjunction in this setting. Then these considerations enable us to compare various positive and negative results on $L^2$ extension from singular hypersurfaces. In particular, we generalize a recent negative result of Guan and Li which places limitations on strengthening such $L^2$ extension by employing a less singular measure in the place of the Ohsawa measure.