A Simple Proof for the Generalized Frankel Conjecture

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1 arxiv: v [math.dg] 30 Jun 2007 A Simple Proof for the Generalized Frankel Conjecture Hui-Ling Gu Department of Mathematics Sun Yat-Sen University Guangzhou, P.R.China Abstract In this short paper, we will give a simple proof for the generalized Frankel conjecture.. Introduction Let M n be an n-dimensional compact Kähler manifold. The famous Frankel conjecture states that: if M has positive holomorphic bisectional curvature, then it is biholomorphic to the complex projective space CP n. This was independently proved by Mori [8] in 979 and Siu-Yau [9] In 980 by using different methods. Mori had got a more general result. His method is to study the deformation of a morphism from CP into the projective manifold M n, while Siu-Yau used the existence result of minimal energy 2-spheres to prove the Frankel conjecture. After the work of Mori and Siu-Yau, it is natural to ask the question for the semi-positive case: what the manifold is if the holomorphic bisectional curvature is nonnegative.

2 This is often called the generalized Frankel conjecture and was proved by Mok [7]. The exact statement is as follows: Theorem. Let (M, h) be an n-dimensional compact Kähler manifold of nonnegative holomorphic bisectional curvature and let ( M, h) be its universal covering space. Then there exist nonnegative integers k, N,, N l and irreducible compact Hermitian symmetric spaces M,, M p of rank 2 such that ( M, h) is isometrically biholomorphic to (C k, g 0 ) (CP N, θ ) (CP N l, θ l ) (M, g ) (M p, g p ) where g 0 denotes the Euclidean metric on C k, g,, g p are canonical metrics on M,, M p, and θ i, i l, is a Kähler metric on CP N i carrying nonnegative holomorphic bisectional curvature. We point out that the three dimensional case of this result was obtained by Bando []. In the special case, for all dimensions, when the curvature operator of M is assumed to be nonnegative, the above result was proved by Cao and Chow [5]. By using the splitting theorem of Howard-Smyth-Wu [6], one can reduce Theorem. to the proof of the following theorem: Theorem.2 Let (M, h) be an n-dimensional compact simply connected Kähler manifold of nonnegative holomorphic bisectional curvature such that the Ricci curvature is positive at one point. Suppose the second Betti number b 2 (M) =. Then either M is biholomorphic to the complex projective space or (M, h) is isometrically biholomorphic to an irreducible compact Hermitian symmetric manifold of rank 2. In [7], Mok proved Theorem.2 and hence the generalized Frankel conjecture. His method depended on Mori s theory of rational curves on Fano manifolds, so it was not completely transcendental in nature. The purpose of this paper is to give a completely transcendental proof of Theorem.2. Our method is inspired by the recent breakthroughs in Ricci flow due to [2, 3, 4]. In [2], by developing a new method constructing the invariant cones to Ricci flow, Böhm and Wilking proved the differentiable sphere theorem for manifolds with positive curvature operator. Recently, Brendle and Schoen [3] proved the 4 - differentiable sphere theorem by using method of [2]. Moreover in [4], the authors 2

3 gave a complete classification of weakly -pinched manifolds. In this paper, we 4 will use the powerful strong maximum principle proposed in [4] to give Theorem.2 a simple proof. Acknowledgement I would like to thank my advisor Professor X.P.Zhu and Professor B.L.Chen for their encouragement, suggestions and discussions. This paper was done under their advice. 2. The Proof of the Main Theorem Proof of the Main Theorem.2. Suppose (M, h) is a compact simply connected Kähler manifold of nonnegative holomorphic bisectional curvature such that the Ricci curvature is positive at one point. We evolve the metric by the Kähler Ricci flow: g t i j(x, t) = R i j(x, t), g i j(x, 0) = h i j(x). According to Bando [], we know that the evolved metric g i j(t), t (0, T), remains Kähler. Then by Proposition. in [7], we know that for t (0, T), g i j(t) has nonnegative holomorphic bisectional curvature and positive holomorphic sectional curvature and positive Ricci curvature everywhere. Suppose (M, h) is not locally symmetric. In the following, we want to show that M is biholomorphic to the complex projective space CP n. Since the smooth limit of locally symmetric space is also locally symmetric, we can obtain that there exists δ (0, T) such that (M, g i j(t)) is not locally symmetric for t (0, δ). Combining the Kählerity of g i j(t) and Berger s holonomy theorem, we know that the holonomy group Hol(g(t)) = U(n). Let P be the fiber bundle whose fiber over p M consists of all orthonormal 2-frames {e α, e β } Tp,0 (M). We define a function u on P (0, δ) by u({e α, e β }, t) = R(e α, e α, e β, e β ) = R αᾱβ β. Clearly we have u 0, since (M, g i j(t)) has nonnegative holomorphic bisectional curvature. Denote F = {u = 0} P (0, δ) of all pairs ({e α, e β }, t) such that 3

4 {e α, e β } has zero holomorphic bisectional curvature with respect to g i j(t). Moreover, we have t R αᾱβ β = R αᾱβ β + R αᾱµ ν R ν µβ β R α µβ ν 2 + R α βµ ν 2. Following Mok [7], we consider the Hermitian form H α (µ, ν) = R αᾱµ ν attached to α and let {e µ } be an orthonormal basis of eigenvectors of H α. In the basis we have µ,ν R αᾱµ ν R ν µβ β = µ R αᾱµ µ R µ µβ β. Also we can arrange the basis {e µ } so that R αᾱ R αᾱ2 2 R αᾱm m > 0 and R αᾱl l = 0 for l > m. First, we claim that: R αᾱµ µ R µ µβ β µ µ,ν R α µβ ν 2 c min{0, inf ξ =,ξ V D2 u(ξ, ξ)}, for some constant c > 0, where V denotes the vertical subspaces. Indeed, we consider the function G χ (ε) = R(α + εχ, α + εχ, β + ε µ C µ e µ, β + ε µ C µ e µ ). Then we have 2 d2 G χ (ε) dε 2 ε=0 = R χ χβ β + µ C µ 2 R αᾱµ µ + 2Re µ C µ R α χβ µ + 2Re µ C µ R α βµ χ. Choose C µ = x µ e iθµ, (µ ), we can obtain that: 2 d2 G χ (ε) dε 2 ε=0 = R χ χβ β + µ x µ 2 R αᾱµ µ + 2 µ x µ Re(e iθµ R α χβ µ + e iθµ R α βµ χ ). Set A µ = R α βµ χ, B µ = R α χβ µ, we have: d2 G χ(ε) 2 dε 2 ε=0 = R χ χβ β + µ x µ 2 R αᾱµ µ + µ x µ (e iθµ B µ + e iθµ B µ + e iθµ A µ + e iθµ A µ ) = R χ χβ β + µ x µ 2 R αᾱµ µ + µ x µ (e iθµ (A µ + B µ ) + e iθµ (A µ + B µ )) 4

5 Choose θ µ such that e iθµ (A µ + B µ ) is real, then we have: 2 d2 G χ (ε) dε 2 ε=0 = R χ χβ β + µ x µ 2 R αᾱµ µ + 2 µ x µ A µ + B µ. If we change α with e iϕ α, then A µ = R α βµ χ is replaced by e iϕ A µ, and B µ = R α χβ µ is replaced by e iϕ B µ, we have: 2 d2 F χ (ε) dε 2 ε=0 = R χ χβ β + µ x µ 2 R αᾱµ µ + 2 µ x µ e iϕ A µ + e iϕ B µ, where F χ (ε) = R(e iϕ α + εχ, e iϕ α + εχ, β + ε µ C µ e µ, β + ε µ C µ e µ ). Choose x µ = eiϕ A µ+e iϕ B µ R αᾱµ µ, for µ m; and x µ = 0, for µ > m. We obtain that then 2 d2 F χ (ε) dε 2 ε=0 = R χ χβ β µ m R αᾱχ χ 2 d2 F χ (ε) dε 2 ε=0 = R αᾱχ χ R χ χβ β A µ 2 + B µ 2 R αᾱµ µ, µ m A µ 2 + B µ 2 R αᾱµ µ R αᾱχ χ. Interchanging the roles of χ and µ, and then taking summation, we have χ m 2R αᾱχ χ R χ χβ β c min{0, inf ξ =,ξ V D 2 u(ξ, ξ)} Hence µ m R αᾱµ µ R µ µβ β + µ m, χ m( A µ 2 + B µ 2 )( Rαᾱχ χ R αᾱµ µ + Rαᾱµ µ R αᾱχ χ ) c min{0, inf ξ =,ξ V D 2 u(ξ, ξ)} + 2 µ m, χ m B µ 2. µ m, ν m R α µβ ν 2 c min{0, inf ξ =,ξ V D2 u(ξ, ξ)}. Moreover, for µ > m, we have R αᾱµ µ = 0. Then by the first variation, we have R α χµ µ = 0, hence by polarization R α χβ µ = 0. Combining these we obtain that R αᾱµ µ R µ µβ β µ µ,ν R α µβ ν 2 c min{0, inf ξ =,ξ V D2 u(ξ, ξ)}, 5

6 for some constant c > 0. Therefore we proved our first claim. By the definition of u and the evolution equation of the holomorphic bisectional curvature, we know that u t = u + αᾱµ ν R ν µβ β R α µβ ν µ,ν(r 2 ) + R α βµ ν 2. µ,ν Combining the above inequality, we obtain that: u t Lu + c min{0, inf ξ =,ξ V D2 u(ξ, ξ)}, where L is the horizontal Laplacian on P, V denotes the vertical subspaces. By Proposition 2 in [4], we know that the set is invariant under parallel transport. F = {u = 0} P (0, δ) Next, we claim that R αᾱβ β > 0 for all t (0, δ). Indeed, suppose not. Then R αᾱβ β = 0 for some t (0, δ). Therefore ({e α, e β }, t) F. And also R αᾱβ β = 0 implies that µ,ν(r αᾱµ ν R ν µβ β R α µβ ν 2 ) = 0, R α βµ ν = 0, µ, ν, R αᾱµᾱ = R β βµ β = 0, µ. We define an orthonormal 2-frames {ẽ α, ẽ β } Tp,0 M by ẽ α = sin θ e α cosθ e β, Then ẽ β = cosθ e α + sin θ e β. ẽ α = sin θ e α cosθ e β, ẽ β = cosθ e α + sin θ e β. 6

7 Since F is invariant under parallel transport and (M, g i j(t)) has holonomy group U(n), we obtain that ({ẽ α, ẽ β }, t) F, that is, On the other hand, R(ẽ α, ẽ α, ẽ β, ẽ β ) = 0. R(ẽ α, ẽ α,ẽ β, ẽ β ) = sin 2 θ cos 2 θr αᾱαᾱ + sin 3 θ cosθr αᾱα β + sin 3 θ cosθr αᾱβᾱ + sin 4 θr αᾱβ β sin θ cos 3 θr α βαᾱ sin 2 θ cos 2 θr α βα β sin 2 θ cos 2 θr α ββᾱ sin 3 θ cosθr α ββ β cos 3 θ sin θr βᾱαᾱ sin 2 θ cos 2 θr βᾱα β sin 2 θ cos 2 θr βᾱβᾱ cosθ sin 3 θr βᾱβ β + cos 4 θr β βαᾱ + cos 3 θ sin θr β βα β + cos 3 θ sin θr β ββᾱ + cos 2 θ sin 2 θr β ββ β = cos 2 θ sin 2 θ(r αᾱαᾱ + R β ββ β). So we have R β ββ β + R αᾱαᾱ = 0, if we choose θ such that cos 2 θ sin 2 θ 0. And this contradicts with the fact that (M, g i j(t)) has positive holomophic sectional curvature. Hence we proved that R αᾱβ β > 0, for all t (0, δ). Therefore by the Frankel conjecture, we know that M is biholomophic to the complex projective space CP n. This completes the proof of Theorem.2. # References [] S. Bando, On three-dimensional compact Kähler manifolds of nonnegative bisectional curvature, J. Diff. Geom. 9, (984), [2] C. Böhm, and B. Wilking, Manifolds with positive curvature operators are space forms, arxiv:math.dg/ June

8 [3] S. Brendle, and R. Schoen, Manifolds with /4-pinched curvature are space forms, arxiv:math.dg/ v2 May [4] S. Brendle, and R. Schoen, Classification of manifolds with weakly /4-pinched curvatures, arxiv:math.dg/ v May [5] H. D. Cao, and B. Chow, Compact Kähler manifolds with nonnegative curvature operator, Invent. Math. 83 (986), [6] A. Howard, B. Smyth, and H. Wu, On compact Kähler manifolds of nonnegative bisectional curvature, I, Acta Math. 47 (98), [7] N. Mok, The uniformization theorem for compact Kähler manifolds of nonnegative bisectional curvature, J. Diff. Geom. 27, (988), [8] S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 0 (979), [9] Y. T. Siu, and S. T. Yau, Complex Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (980),