Transverse kähler geometry of sasaki manifolds and toric sasaki-einstein manifolds

Abstract

In this paper we study compact Sasaki manifolds in view of transverse Kæhler geometry and extend some results in Kæhler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kæhler metric with harmonic Chern forms. The integral invariant f1 for the first Chern class case becomes an obstruction to the existence of transverse Kæhler metric of constant scalar curvature. We prove the existence of transverse Kæhler-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. © 2009 J. differential geometry.