Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
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DOI：
10.4310/jdg/1264601036
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发表时间：
2006-07
影响因子：
2.5
通讯作者：
A. Futaki;Hajime Ono;Guofang Wang
中科院分区：
文献类型：
--
作者：
A. Futaki;Hajime Ono;Guofang Wang
本文从横截K ahler几何的角度研究紧Sasaki流形，并将K ahler几何中的一些结果推广到Sasaki流形.特别地，我们定义了阻碍具有调和Chern形式的横截K“ahler度量存在的积分不变量.第一类情形的积分不变量f1成为常数量曲率横截K ahler度量存在的障碍.证明了紧致复曲面Sasaki流形上的横Kahler-Ricci孤子（或{\it Sasaki-Ricci孤子}）的存在性，其中Reeb叶状的法丛的基本第一Chern形式是正的，接触丛的第一Chern类是平凡的.我们将进一步证明，如果$S$是一个紧凑的环面Sasaki流形与上述假设，然后通过变形的Reeb场，我们得到一个Sasaki-爱因斯坦结构的$S$。作为应用，我们得到了复射影平面两点爆破的正则丛的幂的单位圆丛上的不规则环面Sasaki-Einstein度量。
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.