Open Access
November 2009 Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
Akito Futaki, Hajime Ono, Guofang Wang
J. Differential Geom. 83(3): 585-636 (November 2009). DOI: 10.4310/jdg/1264601036


In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f1$ for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.


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Akito Futaki. Hajime Ono. Guofang Wang. "Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds." J. Differential Geom. 83 (3) 585 - 636, November 2009.


Published: November 2009
First available in Project Euclid: 27 January 2010

zbMATH: 1188.53042
MathSciNet: MR2581358
Digital Object Identifier: 10.4310/jdg/1264601036

Rights: Copyright © 2009 Lehigh University

Vol.83 • No. 3 • November 2009
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