Transverse Kähler Geometry of Sasaki Manifolds and Toric Sasaki-einstein Manifolds

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 in particular the existence of Sasaki-Einstein metrics for compact toric Sasaki manifolds with vanishing f1. We will further show that if S is a compact toric Sasaki manifold such that the basic first Chern class is positive then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain irregular toric Sasaki-Einstein metrics on the unit circle bundles of the powers of the anticanonical bundle of the two-point blow-up of the complex projective plane.