Invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties

Let G be a simply connected semisimple compact Lie group, let X be a simply connected compact K & auml;hler manifold homogeneous under G, and let L be a negative holomorphic line bundle over X. We prove that all G-invariant K & auml;hler metrics on the total space of L arise from the Calabi ansatz. Using this, we show that there exists a unique G-invariant scalar-flat K & auml;hler metric in each G-invariant K & auml;hler class of L. The G-invariant scalar-flat K & auml;hler metrics are automatically asymptotically conical.