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Sasaki–Einstein metrics and K–stability

Tristan C Collins and Gábor Székelyhidi

Geometry & Topology 23 (2019) 1339–1413

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.

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K–stability, Kähler–Einstein, Sasaki
Mathematical Subject Classification 2010
Primary: 32Q20, 53C25
Secondary: 32Q26
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
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