Transverse K'ahler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

摘要

In this paper we study compact Sasaki manifolds in view of transverseK\"ahler geometry and extend some results in K\"ahler geometry to Sasakimanifolds. In particular we define integral invariants which obstruct theexistence of transverse K\"ahler metric with harmonic Chern forms. The integralinvariant $f_1$ for the first Chern class case becomes an obstruction to theexistence of transverse K\"ahler metric of constant scalar curvature. We provethe existence of transverse K\"ahler-Ricci solitons (or {\it Sasaki-Riccisoliton}) on compact toric Sasaki manifolds whose basic first Chern form of thenormal bundle of the Reeb foliation is positive and the first Chern class ofthe contact bundle is trivial. We will further show that if $S$ is a compacttoric Sasaki manifold with the above assumption then by deforming the Reebfield we get a Sasaki-Einstein structure on $S$. As an application we obtainirregular toric Sasaki-Einstein metrics on the unit circle bundles of thepowers of the canonical bundle of the two-point blow-up of the complexprojective plane.