Frankel conjecture and Sasaki geometry

Weiyong He SONG SUN

Differential Geometry mathscidoc:1912.43456

Advances in Mathematics, 291, 912-960, 2016.3
We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Khler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and SiuYau 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 Khler cone. First, we deform the metric along the SasakiRicci flow and obtain a limit SasakiRicci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the SasakiRicci soliton to a SasakiEinstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats
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@inproceedings{weiyong2016frankel,
  title={Frankel conjecture and Sasaki geometry},
  author={Weiyong He, and SONG SUN},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114543602357016},
  booktitle={Advances in Mathematics},
  volume={291},
  pages={912-960},
  year={2016},
}
Weiyong He, and SONG SUN. Frankel conjecture and Sasaki geometry. 2016. Vol. 291. In Advances in Mathematics. pp.912-960. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114543602357016.
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