On K\"ahler manifolds with positive orthogonal bisectional curvature

TL;DR

This paper investigates K"ahler manifolds with positive orthogonal bisectional curvature, showing such manifolds with positive first Chern class are biholomorphic to complex projective space, extending classical results like the Frankel conjecture.

Contribution

It proves that irreducible K"ahler manifolds with preserved positive orthogonal bisectional curvature under Ricci flow are biholomorphic to complex projective space, generalizing previous theorems.

Findings

Any such manifold is biholomorphic to ^n.

Positivity of orthogonal bisectional curvature is preserved under Ricci flow.

Extension of Frankel conjecture to this curvature condition.

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.