Frankel conjecture and Sasaki geometry
Frankel conjecture and Sasaki geometry
复制标题
DOI：
10.1016/j.aim.2015.11.053
复制
发表时间：
2012-02
期刊：
影响因子：
--
通讯作者：
Weiyong He;Song Sun
中科院分区：
文献类型：
--
作者：
Weiyong He;Song Sun
我们对具有正横二分曲率的2n+1维单连通紧致Sasaki流形进行了分类。特别地，对应于这种流形的Kähler锥一定是Cn+1\{0}的双全纯的。作为一个应用，我们恢复了Mori和Siu-Yau关于Frankel猜想的定理，并将其推广到某些orbilold形式。其主要思想是分两步将这种Sasaki流形变形为标准的圆形球体，两步都将复杂结构固定在Kähler锥上。首先，我们沿着Sasaki-Ricci流对度规进行形变，得到了一个具有正横向平分曲率的极限Sasaki-Ricci孤子。然后通过改变Reeb矢量场，本质上降低了体积泛函，我们将Sasaki-Ricci孤子形变为具有正横向对分曲率的Sasaki-Einstein度规，即一个圆球。第二种变形只有在同时处理正则流形和不规则流形时才有可能，即使一开始的流形是正则的(准正则的)，即Kähler流形(或分支流形)。
We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and Siu–Yau on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the Kähler cone. First, we deform the metric along the Sasaki–Ricci flow and obtain a limit Sasaki–Ricci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the Sasaki–Ricci soliton to a Sasaki–Einstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats simultaneously regular and irregular Sasaki manifolds, even if the manifold one starts with is regular (quasi-regular), ie Kähler manifolds (orbifolds).