UC Irvine

This dissertation discusses the Frankel conjecture and the Kähler-Ricci flow approach to it. Frankel conjecture (first proved by Mori and Siu-Yau independently) states that every compact Kähler manifold of positive bisectional curvature is biholomorphic to the complex projective space. On the other hand, the Ricci flow introduced by Hamilton was used by Bando and Mok to generalize Siu-Yau's theorem to nonnegative bisectional curvature case. It's natural to ask if there is a primarily flow proof of the original Frankel conjecture.

The convergence of Kähler-Ricci flow on compact manifolds with positive bisectional curvature would imply such a proof, however not yet been completed. The advances closest to this target might be a series of papers by Phong-Song-Sturm-Weinkove along with the improvements by Cao-Zhu and Zhang. We are going to survey their works in this thesis, and also cover some new result proved by the author.

This thesis is organized as follows. In chapter one, we first collect some fundamental facts of Kähler geometry, and then go over the convergence theory of Kähler-Ricci flow on Fano manifolds built on stability conditions. In chapter two, we review known results on bisectional curvature, and then relate the curvature to the former stability conditions. Finally we will state and prove our new result.