Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L2-cohomology classes

TL;DR

This paper develops an adjunction formula for the Grauert-Riemenschneider canonical sheaf of singular hypersurfaces and applies it to extend L2-cohomology classes from the hypersurface to the ambient manifold, generalizing extension theorems.

Contribution

It introduces a new adjunction formula for the Grauert-Riemenschneider sheaf and demonstrates its use in extending L2-cohomology classes via invariance under bimeromorphic modifications.

Findings

Derived an adjunction formula for the Grauert-Riemenschneider sheaf.

Proved invariance of the L2-extension problem under bimeromorphic modifications.

Reduced the extension problem to the smooth case using embedded resolution.

Abstract

In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface V in a complex manifold M. This adjunction formula is used to study the problem of extending L2-cohomology classes of dbar-closed forms from the singular hypersurface V to the manifold M in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the L2-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of V in M. The smooth case has recently been studied by Berndtsson.