Globally conformally K\"ahler Einstein metrics on certain holomorphic bundles

TL;DR

This paper constructs explicit complete conformally K"ahler Einstein metrics on certain holomorphic bundles using ODE methods, expanding known examples of such metrics on complex manifolds.

Contribution

It introduces new explicit constructions of complete Einstein metrics on holomorphic vector bundles via the Calabi ansatz and its generalizations.

Findings

Existence of non-trivial complete conformally K"ahler Einstein metrics on specific bundles.

Explicit metric expressions in special cases.

Examples of complete metrics on complements of subvarieties in compact K"ahler manifolds.

Abstract

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally K\"ahler Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete K\"ahler-Einstein manifolds. In special cases, we give the explicit expressions of of these metrics. These examples show that there is a compact K\"ahler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally K\"ahler Einstein metric on $M-N$.