arXiv 1202.25892012-02math.DG

Frankel猜想与Sasaki几何

Frankel Conjecture and Sasaki geometry

Weiyong He, Song Sun

备注：: updated version

摘要：

我们对具有正横截对分曲率的$2n+1$维单连通紧致Sasaki流形进行了分类.特别地，对应于这种流形的凯勒锥必须是对$\C^{n+1}\backslash \{0\}$的双全纯。作为应用，我们恢复了关于Frankel猜想的Mori-Siu-Yau定理，并将其推广到某些轨道形式。其主要思想是通过两个步骤将Sasaki流形变形为标准的圆形球面，这两个步骤都将复结构固定在Kähler锥上。首先，我们沿着Sasaki-里奇流对度规进行变形，得到一个具有正横截曲率的极限Sasaki-Ricci孤子。然后通过沿体积泛函的负梯度改变Reeb向量场，将Sasaki-Ricci孤子变形为具有正横截曲率的Sasaki-Einstein度规，即一个圆球.第二种变形只有在同时处理正则和不正则的Sasaki流形时才是可能的，即使一开始的流形是正则的（拟正则的），即Kähler流形（orbifolds）。

英文摘要：

We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem 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 along the negative gradient of the volume functional, we deform the Sasaki-Ricci soliton to a Sasaki-Einstein metric with positive transverse bisectional curvature, i.e. 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), i.e. Kähler manifolds(orbifolds).

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