Kähler geometry on total spaces of vector bundles over elliptic curves
Hanyu Wu, Bo Yang
TL;DR
The paper studies function theory and Kähler geometry on total spaces of vector bundles over an elliptic curve, proving rigidity results that link biholomorphisms of total spaces to bundle isomorphisms and base curves. It develops Calabi-type complete metrics to control holomorphic function growth, establishing a minimal Hadamard order of $2$ and a polynomial-growth framework, and it classifies line-bundle total spaces with nonnegative bisectional curvature, showing this occurs precisely when the Appel–Humbert form vanishes. For rank-2 degree-zero bundles, it furnishes a trichotomy into Type I–III and provides a complete description of holomorphic functions on the total spaces, including when nonconstant functions exist (only in the rational monodromy case for Type III). The paper further constructs complete Hermitian (Gauduchon) metrics with zero Chern-Ricci curvature in Type II and III and studies nonzero-degree cases, including holomorphic convexity results for certain indecomposable bundles, highlighting the rich interplay between algebraic data of the bundle and analytic geometry on the total space.
Abstract
We study function theory and Kähler geometry on total spaces of vector bundles on an elliptic curve. For rank two vector bundles of degree zero, we show that any two total spaces are biholomorphic if and only if the corresponding vector bundles are isomorphic. We also construct complete Gauduchon Hermitian metrics with flat Chern-Ricci curvature on these total spaces. These metrics are natural in the sense that the corresponding spaces of holomorphic functions of polynomial growth coincide with `polynomials' on these spaces. Moreover, we characterize all complete Kähler metrics with nonnegative bisectional curvature on total spaces of line bundles over an elliptic curve.