Invariant scalar-flat K\"ahler metrics on line bundles over generalized flag varieties

TL;DR

This paper demonstrates that all invariant K"ahler metrics on line bundles over homogeneous spaces can be constructed via the Calabi ansatz, and establishes the uniqueness of scalar-flat K"ahler metrics in each K"ahler class.

Contribution

It proves that all invariant K"ahler metrics on such line bundles originate from the Calabi ansatz and confirms the uniqueness of scalar-flat K"ahler metrics in each K"ahler class.

Findings

All invariant K"ahler metrics are from the Calabi ansatz.

Existence of a unique scalar-flat K"ahler metric per K"ahler class.

Characterization of scalar-flat K"ahler metrics on line bundles over flag varieties.

Abstract

Let $G$ be a simply-connected semisimple compact Lie group, $X$ a compact K\"ahler manifold homogeneous under $G$, and $L$ a negative $G$-equivariant holomorphic line bundle over $X$. We prove that all $G$-invariant K\"ahler metrics on the total space of $L$ arise from the Calabi ansatz. Using this, we then show that there exists a unique $G$-invariant scalar-flat K\"ahler metric in each K\"ahler class of $L$.