Negatively curved Kähler metrics on total spaces of a class of vector bundles
Hanyu Wu, Bo Yang
TL;DR
The paper develops a comprehensive framework for complete Kähler metrics with negative holomorphic bisectional curvature on total spaces of vector bundles over compact Kähler bases, via Calabi-type constructions linked to a monotone function χ. It proves a bijective correspondence between BI<0 metrics and decreasing χ, enabling explicit metric realizations and Euclidean-volume-growth cases, and derives sharp dimension bounds for holomorphic functions of polynomial growth in terms of cohomology groups on the base. Using Hessian comparison, three-circle inequalities, and Li-Tam Green-function techniques, the authors establish matching upper and lower bounds for dim O_d(X, g̃), revealing a close connection between fiber growth and base cohomology; equality cases yield rigidity with χ ≡ 0. The results yield Liouville-type theorems for holomorphic mappings, rigidity phenomena for automorphism groups, and a suite of examples over ball quotients and complete intersections, significantly advancing function theory on noncompact negatively curved Kähler manifolds.
Abstract
In this paper we show an abundance of complete Kähler metrics with negative holomorphic bisectional curvature on total spaces of certain vector bundles. Assume that such total spaces are endowed with a wider class of nonpositively curved Kähler metrics. We prove dimension estimates on holomorphic functions on these manifolds, as well as Liouville theorems for holomorphic mappings between them.