Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties

TL;DR

This paper surveys recent advances in L2 extension theorems for holomorphic functions and cohomology classes, especially from non-reduced subvarieties, using techniques like L2 approximation and Bochner-Kodaira methods.

Contribution

It generalizes the Ohsawa-Takegoshi extension theorem to non-reduced subvarieties defined by multiplier ideal sheaves, introducing new techniques for existence proofs.

Findings

Surjectivity holds for restriction to non-reduced subvarieties

Extension results are valid under optimal curvature conditions

L2 approximation is key to the new approach

Abstract

The goal of this survey is to describe some recent results concerning the L 2 extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of L 2 approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) L 2 estimates for the extension in the case of non reduced subvarieties -- the case when Y has singularities or several irreducible components is also a substantial issue.