2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72969This 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 $\bbC^n\setminus\{0\}$. Using this framework we (re)construct a scalar-flat Kähler metric on the blow-up of $\bbC^n$ at the origin from data on the moment polytope.8 pages. Part of submitted versionDifferential Geometrytext