\(L^{2}\)-extension of \(\overline{\partial}\)-closed forms (Q372148)

In the first part of the paper the author proves the following theorem. Let \(X\) be an \(n\)-dimensional compact complex manifold, let \(\Delta\) be a smooth divisor in \(X\), and let \(L\) be a holomorphic line bundle over \(X\). Assume that \(u\) is a smooth \(\overline\partial\)-closed \(L\)-valued \((0,q)\)-form on \(\Delta\) (\(q\geq1\)). Then \(u\) is exact on \(\Delta\) iff for any \(\varepsilon>0\) there exists a \(\overline\partial\)-closed extension \(U\) of \(u\) to \(X\) with \(L^2\)-norm smaller than \(\varepsilon\).NEWLINENEWLINEIn the second part of the paper under review the author presents an alternative proof of \textit{V.~Koziarz}'s theorem [Manuscr. Math. 134, No. 1--2, 43--58 (2011; Zbl 1209.32009)] based on methods from his paper [Ann. Inst. Fourier 46, No. 4, 1083--1094 (1996; Zbl 0853.32024)].