aa r X i v : . [ m a t h . DG ] A ug A Simple Proof for the GeneralizedFrankel Conjecture
Hui-Ling Gu
Department of MathematicsSun Yat-Sen UniversityGuangzhou, P.R.China
Abstract
In this short paper, we will give a simple and transcendental proof forMok’s theorem of the generalized Frankel conjecture. This work is based on themaximum principle in [4] proposed by Brendle and Schoen.
1. Introduction
Let M n be an n -dimensional compact K¨ a hler manifold. The famous Frankelconjecture states that: if M has positive holomorphic bisectional curvature, thenit is biholomorphic to the complex projective space CP n . This was independentlyproved by Mori [9] in 1979 and Siu-Yau [10] in 1980 by using different methods.Mori had got a more general result. His method is to study the deformation ofa morphism from CP into the projective manifold M n , while Siu-Yau used theexistence result of minimal energy 2-spheres to prove the Frankel conjecture. Afterthe work of Mori and Siu-Yau, it is natural to ask the question for the semi-positivecase: what the manifold is if the holomorphic bisectional curvature is nonnegative.1his is often called the generalized Frankel conjecture and was proved by Mok [8].The exact statement is as follows: Theorem 1.1
Let ( M, h ) be an n -dimensional compact K ¨ a hler manifold of non-negative holomorphic bisectional curvature and let ( ˜ M , ˜ h ) be its universal coveringspace. Then there exist nonnegative integers k, N , · · · , N l and irreducible compactHermitian symmetric spaces M , · · · , M p of rank ≥ such that ( ˜ M , ˜ h ) is isomet-rically biholomorphic to ( C k , g ) × ( CP N , θ ) × · · · × ( CP N l , θ l ) × ( M , g ) × · · · × ( M p , g p ) where g denotes the Euclidean metric on C k , g , · · · , g p are canonical metrics on M , · · · , M p , and θ i , ≤ i ≤ l , is a K ¨ a hler metric on CP N i carrying nonnegativeholomorphic bisectional curvature. We point out that the three dimensional case of this result was obtained byBando [1]. 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 [7], one can reduce The-orem 1.1 to the proof of the following theorem: Theorem 1.2
Let ( M, h ) be an n -dimensional compact simply connected K ¨ a hlermanifold of nonnegative holomorphic bisectional curvature such that the Ricci cur-vature is positive at one point. Suppose the second Betti number b ( M ) = 1 . Theneither M is biholomorphic to the complex projective space or ( M, h ) is isometri-cally biholomorphic to an irreducible compact Hermitian symmetric manifold ofrank ≥ . In [8], Mok proved Theorem 1.2 and hence the generalized Frankel conjecture.His method depended on Mori’s theory of rational curves on Fano manifolds, so itwas not completely transcendental in nature. The purpose of this paper is to givea completely transcendental proof of Theorem 1.2.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 Ricciflow, B¨ o hm and Wilking proved the differentiable sphere theorem for manifoldswith positive curvature operator. Recently, Brendle and Schoen [3] proved the -differentiable sphere theorem by using method of [2]. Moreover in [4], the authors2ave a complete classification of weakly -pinched manifolds. In this paper, wewill use the powerful strong maximum principle proposed in [4] to give Theorem1.2 a simple proof. Acknowledgement
I would like to thank my advisor Professor X.P.Zhu and Pro-fessor B.L.Chen for their encouragement, suggestions and discussions. This paperwas done under their advice.
2. The Proof of the Main Theorem
Proof of the Main Theorem 1.2.
Suppose (
M, h ) is a compact simply connectedK¨ a hler manifold of nonnegative holomorphic bisectional curvature such that theRicci curvature is positive at one point. We evolve the metric by the K¨ a hler Ricciflow: ∂∂t g i ¯ j ( x, t ) = − R i ¯ j ( x, t ) ,g i ¯ j ( x,
0) = h i ¯ j ( x ) . According to Bando [1], we know that the evolved metric g i ¯ j ( t ) , t ∈ (0 , T ), re-mains K¨ a hler. Then by Proposition 1.1 in [8], we know that for t ∈ (0 , T ), g i ¯ j ( t )has nonnegative holomorphic bisectional curvature and positive holomorphic sec-tional curvature and positive Ricci curvature everywhere. Moreover, according toHamilton [6], under the evolving orthonormal frame { e α } , we have ∂∂t R α ¯ αβ ¯ β = △ R α ¯ αβ ¯ β + Σ µ,ν ( R α ¯ αµ ¯ ν R ν ¯ µβ ¯ β − | R α ¯ µβ ¯ ν | + | R α ¯ βµ ¯ ν | ) . Suppose (
M, h ) is not locally symmetric. In the following, we want to showthat M is biholomorphic to the complex projective space CP n .Since the smooth limit of locally symmetric space is also locally symmetric, wecan obtain that there exists δ ∈ (0 , T ) such that ( M, g i ¯ j ( t )) is not locally symmetricfor t ∈ (0 , δ ). Combining the K¨ a hlerity of g i ¯ j ( t ) and Berger’s holonomy theorem,we know that the holonomy group Hol( g ( t )) = U ( n ).Let P = S p ∈ M ( T , p ( M ) × T , p ( M )) be the fiber bundle with the fixed metric h and the fiber over p ∈ M consists of all 2-vectors { X, Y } ⊂ T , p ( M ). We define afunction u on P × (0 , δ ) by u ( { X, Y } , t ) = R ( X, X, Y, Y ) , R denotes the pull-back of the curvature tensor of g i ¯ j ( t ). Clearly we have u ≥
0, since (
M, g i ¯ j ( t )) has nonnegative holomorphic bisectional curvature. Denote F = { ( { X, Y } , t ) | u ( { X, Y } , t ) = 0 , X = 0 , Y = 0 } ⊂ P × (0 , δ ) of all pairs ( { X, Y } , t )such that { X, Y } has zero holomorphic bisectional curvature with respect to g i ¯ j ( t ).Following Mok [8], we consider the Hermitian form H α ( X, Y ) = R ( e α , e α , X, Y ),for all X, Y ∈ T , p ( M ) and all p ∈ M , attached to e α . Let { E µ } be an orthonormalbasis associated to eigenvectors of H α . In the basis we have X µ,ν R α ¯ αµ ¯ ν R ν ¯ µβ ¯ β = X µ R ( e α , e α , E µ , E µ ) R ( E µ , E µ , e β , e β ) , and X µ,ν | R α ¯ µβ ¯ ν | = X µ,ν | R ( e α , E µ , e β , E ν ) | . First, we claim that: X µ,ν R α ¯ αµ ¯ ν R ν ¯ µβ ¯ β − X µ,ν | R α ¯ µβ ¯ ν | ≥ c · min { , inf | ξ | =1 ,ξ ∈ V D u ( { e α , e β } , t )( ξ, ξ ) } , for some constant c >
0, where V denotes the vertical subspaces.Indeed, inspired by Mok [8], for any given ε > χ ∈ { , , ··· , n } ,we consider the function e G χ ( ε ) = ( R + ε R )( e α + εE χ , e α + εE χ , e β + ε X µ C µ E µ , e β + ε X µ C µ E µ ) , where R is a curvature operator defined by ( R ) i ¯ jk ¯ l = g i ¯ j g k ¯ l + g i ¯ l g k ¯ j and C µ arecomplex constants to be determined later. For the simplicity, we denote e R = R + ε R , then e G χ ( ε ) = e R ( e α + εE χ , e α + εE χ , e β + ε X µ C µ E µ , e β + ε X µ C µ E µ ) . Then a direct computation gives · d e G χ ( ε ) dε | ε =0 = e R ( E χ , E χ , e β , e β ) + P µ | C µ | e R ( e α , e α , E µ , E µ )+2 Re P µ C µ e R ( e α , E χ , e β , E µ ) + 2 Re P µ C µ e R ( e α , e β , E µ , E χ ) . C µ = x µ e iθ µ , ( µ ≥ x µ , θ µ are constants to be determined later, theabove identity is: · d e G χ ( ε ) dε | ε =0 = e R ( E χ , E χ , e β , e β ) + P µ | x µ | e R ( e α , e α , E µ , E µ )+2 P µ x µ · Re ( e − iθ µ e R ( e α , E χ , e β , E µ ) + e iθ µ e R ( e α , e β , E µ , E χ )) . Following Mok [8], by setting A µ = e R ( e α , e β , E µ , E χ ) , B µ = e R ( e α , E χ , e β , E µ ), wehave: · d e G χ ( ε ) dε | ε =0 = e R ( E χ , E χ , e β , e β ) + P µ | x µ | e R ( e α , e α , E µ , E µ )+ P µ x µ ( e − iθ µ B µ + e iθ µ B µ + e iθ µ A µ + e − iθ µ A µ )= e R ( E χ , E χ , e β , e β ) + P µ | x µ | e R ( e α , e α , E µ , E µ )+ P µ x µ · ( e iθ µ ( A µ + B µ ) + e iθ µ ( A µ + B µ ))By choosing θ µ such that e iθ µ ( A µ + B µ ) is real and positive, the identity becomes: · d e G χ ( ε ) dε | ε =0 = e R ( E χ , E χ , e β , e β ) + P µ | x µ | e R ( e α , e α , E µ , E µ )+2 P µ x µ · | A µ + B µ | . If we change e α with e iϕ e α , then A µ = e R ( e α , e β , E µ , E χ ) is replaced by e iϕ A µ , and B µ = e R ( e α , E χ , e β , E µ ) is replaced by e − iϕ B µ , we have: · d e F χ ( ε ) dε | ε =0 = e R ( E χ , E χ , e β , e β ) + P µ | x µ | e R ( e α , e α , E µ , E µ )+2 P µ x µ · | e iϕ A µ + e − iϕ B µ | , where e F χ ( ε ) = e R ( e iϕ e α + εE χ , e iϕ e α + εE χ , e β + ε X µ C µ E µ , e β + ε X µ C µ E µ ) . Since the curvature operators R and R have nonnegative and positive holomorphicbisectional curvature respectively, we know that the operator e R = R + ε R haspositive holomorphic bisectional curvature. Now by choosing x µ = − | e iϕ A µ + e − iϕ B µ | e R ( e α ,e α ,E µ ,E µ ) ,for µ ≥
1, it follows that12 π Z π ( 12 · d e F χ ( ε ) dε | ε =0 ) dϕ = e R ( E χ , E χ , e β , e β ) − X µ | A µ | + | B µ | e R ( e α , e α , E µ , E µ )5nd then e R ( e α , e α , E χ , E χ ) · π R π ( · d e F χ ( ε ) dε | ε =0 ) dϕ = e R ( e α , e α , E χ , E χ ) e R ( E χ , E χ , e β , e β ) − P µ | A µ | + | B µ | e R ( e α ,e α ,E µ ,E µ ) e R ( e α , e α , E χ , E χ ) . Note that e F χ ( ε ) = e R ( e iϕ e α + εE χ , e iϕ e α + εE χ , e β + ε P µ C µ E µ , e β + ε P µ C µ E µ )= e R ( e α + εe − iϕ E χ , e α + εe − iϕ E χ , e β + ε P µ C µ E µ , e β + ε P µ C µ E µ ) . Interchanging the roles of E χ and E µ , and then taking summation, we have P χ e R ( e α , e α , E χ , E χ ) e R ( E χ , E χ , e β , e β ) ≥ c · min { , inf | ξ | =1 ,ξ ∈ V D e u ( { e α , e β } , t )( ξ, ξ ) } + P µ,χ ( | A µ | + | B µ | )( e R ( e α ,e α ,E χ ,E χ ) R ( e α ,e α ,E µ ,E µ ) + e R ( e α ,e α ,E µ ,E µ ) e R ( e α ,e α ,E χ ,E χ ) ) ≥ c · min { , inf | ξ | =1 ,ξ ∈ V D e u ( { e α , e β } , t )( ξ, ξ ) } + 2 P µ,χ | e R ( e α , E χ , e β , E µ ) | , where e u ( { X, Y } , t ) = e R ( X, X, Y, Y ) = R ( X, X, Y, Y ) + ε R ( X, X, Y, Y ) and c isa positive constant that does not depend on ε .Hence P µ e R ( e α , e α , E µ , E µ ) e R ( E µ , E µ , e β , e β ) − P µ,ν | e R ( e α , E µ , e β , E ν ) | ≥ c · min { , inf | ξ | =1 ,ξ ∈ V D e u ( { e α , e β } , t )( ξ, ξ ) } . Since ε > ε →
0, then we obtain that: X µ,ν R α ¯ αµ ¯ ν R ν ¯ µβ ¯ β − X µ,ν | R α ¯ µβ ¯ ν | ≥ c · min { , inf | ξ | =1 ,ξ ∈ V D u ( { e α , e β } , t )( ξ, ξ ) } , for some constant c >
0. Therefore we proved our first claim.By the definition of u and the evolution equation of the holomorphic bisectionalcurvature, we know that ∂∂t u ( { X, Y } , t ) = △ u ( { X, Y } , t ) + P µ,ν R ( X, X, e µ , e ν ) R ( e ν , e µ , Y, Y ) − P µ,ν | R ( X, e µ , Y, e ν ) | + P µ,ν | R ( X, Y , e µ , e ν ) | . ∂u∂t ≥ Lu + c · min { , inf | ξ | =1 ,ξ ∈ V D u ( ξ, ξ ) } , where L is the horizontal Laplacian on P , V denotes the vertical subspaces. ByProposition 2 in [4], (Actually, the same argument still holds for the bundle P in[4] changed by the bundle P defined in our paper.), we know that the set F = { ( { X, Y } , t ) | u ( { X, Y } , t ) = 0 , X = 0 , Y = 0 } ⊂ P × (0 , δ )is invariant under parallel transport.Next, we claim that R α ¯ αβ ¯ β > t ∈ (0 , δ ) . Indeed, suppose not. Then R α ¯ αβ ¯ β = 0 for some t ∈ (0 , δ ). Therefore( { e α , e β } , t ) ∈ F. Combining R α ¯ αβ ¯ β = 0 and the evolution equation of the curvature operator andthe first variation, we can obtain that: P µ,ν ( R α ¯ αµ ¯ ν R ν ¯ µβ ¯ β − | R α ¯ µβ ¯ ν | ) = 0 ,R α ¯ βµ ¯ ν = 0 , ∀ µ, ν,R α ¯ αµ ¯ β = R β ¯ βµ ¯ α = 0 , ∀ µ. We define an orthonormal 2-frames { f e α , f e β } ⊂ T , p ( M ) by f e α = sin θ · e α − cos θ · e β , f e β = cos θ · e α + sin θ · e β . Then f e α = sin θ · e α − cos θ · e β , f e β = cos θ · e α + sin θ · e β . Since F is invariant under parallel transport and ( M, g i ¯ j ( t )) has holonomy group U ( n ), we obtain that ( { f e α , f e β } , t ) ∈ F, that is, R ( f e α , f e α , f e β , f e β ) = 0 .
7n the other hand, R ( f e α , f e α , f e β , f e β ) = sin θ cos θR α ¯ αα ¯ α + sin θ cos θR α ¯ αα ¯ β + sin θ cos θR α ¯ αβ ¯ α + sin θR α ¯ αβ ¯ β − sin θ cos θR α ¯ βα ¯ α − sin θ cos θR α ¯ βα ¯ β − sin θ cos θR α ¯ ββ ¯ α − sin θ cos θR α ¯ ββ ¯ β − cos θ sin θR β ¯ αα ¯ α − sin θ cos θR β ¯ αα ¯ β − sin θ cos θR β ¯ αβ ¯ α − cos θ sin θR β ¯ αβ ¯ β + cos θR β ¯ βα ¯ α + cos θ sin θR β ¯ βα ¯ β + cos θ sin θR β ¯ ββ ¯ α + cos θ sin θR β ¯ ββ ¯ β = cos θ sin θ ( R α ¯ αα ¯ α + R β ¯ ββ ¯ β ) . So we have R β ¯ ββ ¯ β + R α ¯ αα ¯ α = 0, if we choose θ such that cos θ sin θ = 0. Andthis contradicts with the fact that ( M, g i ¯ j ( t )) has positive holomorphic sectionalcurvature. Hence we proved that R α ¯ αβ ¯ β >
0, for all t ∈ (0 , δ ).Therefore by the Frankel conjecture, we know that M is biholomorphic to thecomplex projective space CP n .This completes the proof of Theorem 1.2. References [1] S. Bando,
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