Optimal $L^2$ Extensions of Openness Type

Abstract

We study the following optimal $L^2$ extension problem of openness type: given a complex manifold $M$, a closed subvariety $S\subset M$ and a holomorphic vector bundle $E\to M$, for any $L^2$ holomorphic section $f$ defined on some open neighborhood $U$ of $S$, find an $L^2$ holomorphic section $F$ on $M$ such that $F{|}_S=f{|}_S$, and the $L^2$ norm of $F$ on $M$ is optimally controlled by the $L^2$ norm of $f$ on $U$. Answering the above problem, we prove an optimal $L^2$ extension theorem of openness type on weakly pseudoconvex Kähler manifolds, which generalizes a couple of known results on such a problem. Moreover, we prove a product property for certain minimal $L^2$ extensions and give an alternative proof to a version of the above $L^2$ extension theorem. We also present some applications to the usual optimal $L^2$ extension problem and the equality part of Suita's conjecture.

72 pages. Improved version of the first author's PhD thesis at AMSS, CAS. In this final version, Sections 7.5 and 7.7 are new. Due to length considerations, this article was divided into two parts during the submission. The first part (Sect. 1-6) was published online by Math. Ann., and the second part (Sect. 7) was accepted by Ann. Inst. Fourier