## Journal of Differential Geometry

### Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

#### Abstract

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.

#### Article information

Source
J. Differential Geom., Volume 83, Number 3 (2009), 585-636.

Dates
First available in Project Euclid: 27 January 2010

https://projecteuclid.org/euclid.jdg/1264601036

Digital Object Identifier
doi:10.4310/jdg/1264601036

Mathematical Reviews number (MathSciNet)
MR2581358

Zentralblatt MATH identifier
1188.53042

#### Citation

Futaki, Akito; Ono, Hajime; Wang, Guofang. Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differential Geom. 83 (2009), no. 3, 585--636. doi:10.4310/jdg/1264601036. https://projecteuclid.org/euclid.jdg/1264601036