Generically surjective morphisms of holomorphic vector bundles via degenerations
Roberto Albesiano
TL;DR
The paper proves an $L^2$-estimate for solving $g=\gamma f$ with $\gamma:F\to G$ a generically surjective holomorphic morphism between vector bundles on a (essentially) Stein manifold, under a curvature inequality mixing $\Theta_{h_F},\Theta_{h_G}$ and the eigenvalue functions $\lambda_{h},\Lambda_{h}$. It introduces a degenerating family of norms, realized by a metric on the hyperplane bundle over $P(F^*\otimes G)$, and shows these norms are convex in the degeneration parameter via a vector-bundle version of Berndtsson's positivity theorem. The proof reduces the $L^2$-estimate to a dual problem and uses a careful curvature analysis, including Demailly–Skoda's Nakano→Griffiths transition, to ensure the necessary positivity. The work generalizes Skoda-type division theorems, recovers special cases with Griffiths-nonnegative curvature and line-bundle twists, and provides a robust degenerative technique for L^2 extension-type problems in complex geometry.
Abstract
We prove an $L^2$ theorem on generically surjective morphism of holomorphic vector bundles via a degeneration argument, generalizing the author's previous work on the $L^2$ division theorem of Skoda. The proof is based on Berndtsson's theorem on the positivity of direct image bundles and is inspired by Berndtsson and Lempert's proof of the $L^2$ extension theorem.