K\"ahler geometry on total spaces of vector bundles over elliptic curves

TL;DR

This paper explores the complex and K"ahler geometry of total spaces of vector bundles over elliptic curves, establishing biholomorphic classifications and constructing special metrics with flat Chern-Ricci curvature.

Contribution

It classifies total spaces of rank two vector bundles over elliptic curves up to biholomorphism and constructs complete Hermitian metrics with flat Chern-Ricci curvature.

Findings

Total spaces of rank two degree-zero bundles are biholomorphic iff bundles are isomorphic.

Constructed complete Gauduchon Hermitian metrics with flat Chern-Ricci curvature.

Characterized all complete K"ahler metrics with nonnegative bisectional curvature on line bundle total spaces.

Abstract

We study function theory and K\"ahler 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 K\"ahler metrics with nonnegative bisectional curvature on total spaces of line bundles over an elliptic curve.