$L^2$ extension of $\bar\partial$-closed forms on weakly pseudoconvex K\"ahler manifolds

TL;DR

This paper extends $L^2$ $ar ext{d}$-closed form extension theorems from Stein to weakly pseudoconvex K"ahler manifolds, broadening the applicability of such extension results under mixed positivity conditions.

Contribution

It generalizes existing $L^2$ extension theorems for $ar ext{d}$-closed forms to a wider class of complex manifolds, incorporating weakly pseudoconvex K"ahler manifolds.

Findings

Extended $L^2$ extension theorems to weakly pseudoconvex K"ahler manifolds.

Established regularity results for modified sections in the extension process.

Demonstrated applicability under mixed positivity conditions.

Abstract

Combining V. Koziarz's observation about the regularity of some modified section related to the initial extension with J. McNeal--D. Varolin's regularity argument, we generalize two theorems of McNeal--Varolin for the $L^2$ extension of $\bar\partial$-closed high-degree forms on a Stein manifold to the weakly pseudoconvex K\"ahler case under mixed positivity conditions.