An application of Guillemin-Abreu theory to a non-abelian group action

Abstract

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin & Abreu to the case of the standard linear $\SU(n)$ action on ${\mathbb{C}}^n\setminus \{0\}$. Using this framework we (re)construct a scalar-flat Kähler metric on the blow-up of ${\mathbb{C}}^n$ at the origin from data on the moment polytope.

8 pages. Part of submitted version

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• A symplectic construction of Calabi's extremal Kähler metrics on the blow-up of $\bbC\bbP^n$ at one point