A New Ricci Flow Proof of Frankel Conjecture

1 Introduction The uniformalization of Kähler manifold has been an important topic in geometry for long. The famous Frankel Conjecture states: Compact n-dimensional Kähler Manifolds of Positive Bisectional Curvature are Bi-holomorphic to P n C. [10] by using stable harmonic maps in the context of Kähler geometry. Actually, Mori proved the Harthshorne Conjecture: Every irreducible n-dimensional nonsingular projective variety with ample tangent bundle defined over an algebraic closed field k of characteristic ≥ 0 is isomorphic to the projective space P n k. Their methods have nothing to do with the deformation of Kähler metrics. In 1982, R.Hamilton introduced the Ricci flow in [6], it is a geometric evolution equation: ∂g ij ∂t = −2Ric ij and in the context of Kähler geometry, the flow can be written as : 1 It has been hoped that the Ricci flow can be used to prove uniformization theorems, in particular, a different proof the Frankel conjecture. In fact, using the Ricci flow, R. Hamilton proved that manifolds admit a metric with positive curvature operator are space forms in dim 3 in [6] and dimension 4 in [7]. He conjectured that the same is true for all dimensions. It is not until Böhm and W ilking in [1] does the conjecture in all dimensions be proved. In the Kähler case, there has been a long history of using the Ricci flow to prove the existence of canonical metrics which leads to a proof of the Frankel conjecture. Hamilton in [9] and Chow in [13] proved that the Ricci flow has a global solution which converges to the standard one on S 2 if the initial metric has positive curvature. Bando proved in dimension 3 in [14] and Mok proved in general dimensions in [15] that the Kähler-Ricci flow preserves the positivity of bisectional curvature. Assuming that the underlying compact Kähler manifold admits a Kähler-Einstein metric, X.X. Chen and G. Tian in [12] proved that the global solution of the Kähler-Ricci flow converges to the Kähler-Einstein metric. For related works on Kähler-Ricci flow without assuming positivity of curvature, see B. Chow [17], H.D. Cao [16], Tian-Zhu [18]. The aim of this paper is to give a proof the Frankel conjecture by using the Kähler-Ricci flow alone without assuming apriori the existence of Kähler-Einstein metrics. Our proof is an extension of the new method Böhm and W ilking developed in [1] to Kähler manifolds. Here …