Extension of L^2, di-bar-closed, forms

TL;DR

This paper proves an extension theorem for di-bar-closed forms from a hyperplane intersection to the entire pseudoconvex domain, providing estimates for the extended form's L^2-norm and addressing gaps in previous proofs.

Contribution

It introduces a new, simpler method to extend di-bar-closed forms with harmonic coefficients, filling gaps in existing literature and extending results beyond holomorphic functions.

Findings

Extension of di-bar-closed forms with harmonic coefficients

L^2-norm estimate for the extended form

Addresses gaps in previous proofs involving singular weights

Abstract

We prove extension of a di-bar-closed, smooth, form from the intersection of a pseudoconvex domain with a complex hyperplane to the whole domain. The extension form is di-bar-closed, has harmonic coefficients and its L^2-norm is estimated by the L^2-norm of the trace. For holomorphic functions this is proved by Ohsawa-Takegoshi [12]. For forms of higher degree, this is stated by Manivel [9]. It seems, however, that the proof contains a gap because of the use of a a singular weight and the failure of regularity for the solution of the related di-bar-equation. There is a rich literature on the subject (cf. among otheres [7], [14]) but it does not seem to contain complete answer to the question. Also, the problem of extending cohomology classes of di-bar of higher degree in a compact Kahler space is addressed in [8] and [3]. Apart from the formal analogy, this has little in common with our problem in which these classes are 0. We also believe, comparing to the preceding literature, that our approach is original and, somewhat, simpler.