Sasaki–Einstein metrics and K–stability

Collins, Tristan; Székelyhidi, Gábor

doi:10.2140/gt.2019.23.1339

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Sasaki–Einstein metrics and K–stability

Tristan C Collins and Gábor Székelyhidi

Geometry & Topology 23 (2019) 1339–1413

DOI: 10.2140/gt.2019.23.1339

Abstract

We show that a polarized affine variety with an isolated singularity admits a Ricci flat Kähler cone metric if and only if it is K–stable. This generalizes the Chen–Donaldson–Sun solution of the Yau–Tian–Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki–Einstein metrics.

Keywords

K–stability, Kähler–Einstein, Sasaki

Mathematical Subject Classification 2010

Primary: 32Q20, 53C25

Secondary: 32Q26

References

Publication

Received: 25 September 2017

Revised: 11 July 2018

Accepted: 8 August 2018

Published: 28 May 2019

Proposed: Simon Donaldson

Seconded: John Lott, Tobias H Colding

Authors

Tristan C Collins
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Gábor Székelyhidi
Department of Mathematics University of Notre Dame Notre Dame, IN United States

Volume 23, issue 3 (2019)