An Extension Theorem in Terms of Adjoint Ideal Sheaves

Abstract

In the context of the study of the Ohsawa–Takegoshi L² extension theorem, a "qualitative" extension theorem in terms of adjoint ideal sheaves (i.e., extension result without L² estimates but with some form of local L² condition ensured) is proved on compact Kähler manifolds via harmonic theory and residue computations. The arguments are adapted from those in the study of the injectivity theorem by Chan–Choi–Matsumura. The result guarantees, in particular, the existence of extensions of line-bundle-valued holomorphic top forms over the union of any log-canonical centres of the same codimension to top forms over the ambient space under suitable positivity assumptions.