On K\"ahler manifolds with positive orthogonal bisectional curvature

Abstract

In this paper, we study any K\"ahler manifold where the positive orthogonal bisectional curvature is preserved on the K\"ahler Ricci flow. Naturally, we always assume that the first Chern class $C_1$ is positive. In particular, we prove that any irreducible K\"ahler manifold with such property must be biholomorphic to $\mathbb{C}\mathbb{P}^n. $ This can be viewed as a generalization of Siu-Yau\cite{Siuy80}, Morri's solution \cite{Mori79} of the Frankel conjecture. According to [8], note that any K\"ahler manifold with 2-positive traceless bisectional curvature operator is preserved under Kahler Ricci flow; which in turns implies the positivity of orthogonal bisectional curvature under the flow.