Calabi-Yau metrics on canonical bundles of flag varieties

Abstract

This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz to general K\"{a}hler classes. We give some examples of explicit families, in particular, a formula for the two dimensional family of asymptotically conical metrics on the canonical bundle of $F_{1,2}$.