un 2 00 6 On Kähler manifolds with positive orthogonal bisectional curvature

The famous Frankel conjecture asserts that any compact Kähler manifold with positive bisectional curvature must be biholomorphic to CP n. This conjecture was settled affirmatively in early 1980s by two groups of mathematicians independently: Siu-Yau[16] via differential geometry method and Morri [15] by algebraic method. There are many interesting papers following this celebrated work; in particular to understand the classification of Kähler manifold with non-negative bisectional curvature, readers are referred to N. Mok's work [14] for further references. In 1982, R. Hamilton [10] introduced the Ricci flow as a means to deform any Riemannian metric in a canonical way to a Einstein metric. He particularly showed that, in any 3-dimensional compact manifold, the positive Ricci curvature is preserved by the Ricci flow. Moreover, the Ricci flow deforms the metric more and more towards Einstein metric. Consequently, he proved that the underlying manifold must be diffeomorphic to S 3 or a finite quotient of S 3. By a theorem of M. Berger in the 1960s, any Kähler Einstein metric with positive bisectional curvature is the Fubni-Study metric (with constant bisectional curvature). A natural and long standing problem for Kähler Ricci flow is: In CP n , is Kähler Ricci flow converges to the Fubni-Study metric if the initial metric has positive bisectional curvature? There are many interesting work in this direction in 1990s (c.f. [1] [14]) and the problem was completely settled in 2000 by [5] [6] affirmatively. One key idea is the introduction of a series new geometrical functionals which play a crucial role in deriving the bound of scalar curvature, diameter and Sobleve constants etc.. It is well known that the positivity of bisectional curvature is preserved along the Kähler Ricci flow, due to S. Bando [1] in dimension 3 and N. Mok in general dimension [14]. Following the work of N. Mok, in an unpublished work of Cao-Hamilton, they claimed that the positive orthogonal bisectional curvature is also preserved under the Kähler Ricci flow. In any Kähler manifold, we can split the space of (1, 1) forms into two orthogonal components: the line spanned by the Kähler form, and its orthogonal complement Λ 1,1 0. The traceless part of the bisectional curvature can be viewed as an operator acting on this subspace Λ 1,1 0. We call a Kähler metric has 2-positive traceless bisectional curvature if the sum of any two eigenvalues of the tracless bisectional curvature operator …