Transverse K\"ahler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

TL;DR

This paper explores the transverse Kähler geometry of Sasaki manifolds, introduces integral invariants as obstructions to certain metrics, and demonstrates the existence of Sasaki-Ricci solitons and Sasaki-Einstein structures on toric Sasaki manifolds.

Contribution

It extends Kähler geometric results to Sasaki manifolds, defines new integral invariants, and constructs Sasaki-Ricci solitons and Sasaki-Einstein metrics on specific toric Sasaki manifolds.

Findings

Existence of transverse Kähler-Ricci solitons on certain toric Sasaki manifolds.

Integral invariants obstruct the existence of specific transverse Kähler metrics.

Construction of irregular toric Sasaki-Einstein metrics on circle bundles over blow-ups of complex projective planes.

Abstract

In this paper we study compact Sasaki manifolds in view of transverse K\"ahler geometry and extend some results in K\"ahler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse K\"ahler metric with harmonic Chern forms. The integral invariant $f_1$ for the first Chern class case becomes an obstruction to the existence of transverse K\"ahler metric of constant scalar curvature. We prove the existence of transverse K\"ahler-Ricci solitons (or {\it 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 irregular toric Sasaki-Einstein metrics on the unit circle bundles of the powers of the canonical bundle of the two-point blow-up of the complex projective plane.