Kähler geometry on total spaces of vector bundles over elliptic curves

2025/11/12 by Han-Yu Wu, Bo Yang, Wu, Hanyu +1
Mathematics · #32Q05 #32Q15 #53C55 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2511.08906

openalex publication_date 2025/11/12 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/28

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.

Citations

• Negatively curved Kähler metrics on total spaces of a class of vector bundles
• Atiyah's work on holomorphic vector bundles and gauge theories
• Comparison geometry of holomorphic bisectional curvature for Kaehler manifolds and limit spaces
• Chern-Ricci flows on noncompact complex manifolds
• On Yau's uniformization conjecture
• On the Steinness of a class of Kähler manifolds
• On the complex structure of Kähler manifolds with nonnegative curvature
• A momentum construction for circle-invariant Kahler metrics
• Vector Bundles Over an Elliptic Curve

Related

• Negatively curved Kähler metrics on total spaces of a class of vector bundles
• Vector Bundles Over an Elliptic Curve
• A natural hermitian metric associated with local universal families of compact Ricci-flat Kähler manifolds
• A note about scalar curvature on the total space of a vector bundle
• t-Gauduchon Ricci-flat metrics on non-Kähler Calabi-Yau manifolds