Toric, U(2), and LeBrun metrics

Abstract

The LeBrun ansatz was designed for scalar-flat K¨ahler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is K¨ahler, scalar-flat, or extremal K¨ahler. Second, toric K¨ahler metrics (such as the generalized Taub-NUTs) and U (2)-invariant metrics (such as the Fubini-Study or Page metrics) are certainly expressible in the LeBrun ansatz. We give general formulas for such transitions. We close the paper with examples, and find expressions for two examples—the exceptional half-plane metric and the Page metric—in terms of the LeBrun ansatz. segundo lugar, m´etricas t´oricas K¨ahler (tales como las Taub-NUT generalizadas) y m´etricas U (2)-invariantes (tales como la m´etrica de Fubini-Study o la de Page) son ciertamente expresables en el ansatz de LeBrun. Damos f´ormulas generales para tales transiciones. Concluimos el art´ıculo con ejemplos, y encontramos expresiones para dos ejemplos—la m´etrica excep-cional del semiplano y la m´etrica de Page—en t´erminos del ansatz de LeBrun.