Positivity preserving along a flow over projective bundle
aa r X i v : . [ m a t h . DG ] J a n POSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVEBUNDLE
XUEYUAN WAN
Abstract.
In this paper, we introduce a flow over the projective bundle p : P ( E ∗ ) → M ,which is a natural generalization of both Hermitian-Yang-Mills flow and K¨ahler-Ricci flow. Weprove that the semipositivity of curvature of the hyperplane line bundle O P ( E ∗ ) (1) is preservedalong this flow under the null eigenvector assumption (see Theorem 1.9). As applications, weprove that the semipositivity is preserved along the this flow if the base manifold M is a curve,which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Millsflow over a curve. And we also reprove that the nonnegativity of holomorphic bisectionalcurvature is preserved under K¨ahler-Ricci flow. Contents
Introduction 11. Preliminaries 41.1. Complex Finsler vector bundle 41.2. Maximum principle for real (1 , Introduction
In the celebrated paper [28], Siu and Yau presented a differential geometric proof of thefamous Frankel conjecture in K¨ahler geometry, which states that a compact K¨ahler manifold M n with positive holomorphic bisectional curvature must be biholomorphic to a complex pro-jective space P n . For a compact K¨ahler manifold with nonnegative holomorphic bisectionalcurvature, Mok [24] proved a generalized Frankel conjecture and obtained a uniformizationtheorem, and H. Gu [16] gave a new proof of Mok’s uniformization theorem. They both usedthe K¨ahler-Ricci flow and considered the variation of holomorphic bisectional curvature alongthis flow. Especially, the nonnegativity of holomorphic bisectional curvature is preserved under K¨ahler-Ricci flow [24, Proposition 1.1]. Later on, there are many references about studyingand generalizing the Frankel conjecture by using K¨ahler-Ricci flow, include [7, 8, 17, 27].In 1979, Mori [25] proved the famous Hartshorne’s conjecture. In case the ground field is C , a compact complex manifold M was proved to biholomorphic to P n if its tangent bundle isample, which implies the Frankel conjecture. A holomorphic vector bundle E is ample in thesense of Hartshorne if and only if the hyperplane line bundle O P ( E ∗ ) (1) is a positive line bundleover P ( E ∗ ) (see [18, Proposition 3.2]), i.e., there is a positive curvature metric on O P ( E ∗ ) (1). If( E, h ) is a Hermitian vector bundle with Griffiths positive curvature (see Definition 1.7), then E is an ample vector bundle. In [15], Griffiths conjectured its converse also holds, namely E canadmit a Hermitian metric with Griffiths positive curvature if E is ample. Both Mori’s theoremand Griffiths conjecture will be proved if one can deform the given positive curvature metric toanother “better” metric with positive curvature, for example, the K¨ahler metric with positiveholomorphic bisectional curvature for Mori’s theorem, and Hermitian metric with Girffithspositive curvature for Griffiths conjecture. Naturally, one wants to define a certain flow overthe projective bundle P ( E ∗ ), such that the positivity of the curvature of O P ( E ∗ ) (1) is preservedunder this flow. This is also the motivation for the author to study the positivity preservingalong the flow (0.1) over projective bundle.Let M be a compact complex manifold of dimension n , and π : E → M be a holomorphicvector bundle of rank r over M . Let E ∗ denote the dual bundle of E , and p : P ( E ∗ ) :=( E ∗ ) o / C ∗ → M denote the projective bundle, where ( E ∗ ) o denotes the set of all the nonzeroelements in E ∗ . For any strongly pseudoconvex complex Finsler metric G on E ∗ (see Definition1.1), there exists the following canonical decomposition (see Remark 1.5, (3)) √− ∂ ¯ ∂ log G = − Ψ + ω F S , which is a curvature form of O P ( E ∗ ) (1) and represents the first Chern class 2 πc ( O P ( E ∗ ) (1)).Here Ψ is called the Kobayashi curvature (see [12, Definition 1.2]), which is given in (1.4), and ω F S is a positive (1 , p : P ( E ∗ ) → M , which is defined in (1.10).According to this decomposition, Kobayashi [19] gave a characterization of ample vector bundle,i.e., E is ample if and only if there exists a strongly pseudoconvex complex Finsler metric on E ∗ such that Ψ < ω ( G ) = p ∗ ω , where ω = √− g α ¯ β ( G ) dz α ∧ d ¯ z β is a K¨ahler metric on M dependingsmoothly on a Finsler metric G . One can define a Hermitian metric on P ( E ∗ ) byΩ := ω ( G ) + ω F S . Let G be a strongly pseudoconvex complex Finsler metric on E ∗ , we consider the followingflow over the projective bundle P ( E ∗ ): ∂∂t log G = ∆ Ω log G,ω
F S > ,G (0) = G . (0.1)Here ∆ Ω := √− ∂ ¯ ∂ , Λ is the adjoint operator of Ω ∧ • . OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 3
One can also define a horizontal and real (1 , T on P ( E ∗ ) as follow,( −√− T ( u, u ) := h R g ( u, u ) , − Ψ i Ω − (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) (0.2)for any horizontal vector u = u α δδz α , where h R g ( u, u ) , − Ψ i Ω := ( − Ψ) α ¯ δ g α ¯ β g γ ¯ δ R gγ ¯ βσ ¯ τ u σ ¯ u τ and (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) := (log G ) a ¯ b ∂ a Ψ α ¯ β ∂ b Ψ γ ¯ τ u α u γ g τ ¯ β , R g denotes the Chern curvature of ω . Nowwe assume that T satisfies the the null eigenvector assumption (see Theorem 1.9), namely( −√− T ( U, U ) ≥ ∂ ¯ ∂ log G ≥ i U ( ∂ ¯ ∂ log G ) = 0 for a (1 , U of T P ( E ∗ ). By using the maximum principle for real (1 , Theorem 0.1.
Let π : ( E ∗ , G ) → M be a holomorphic Finsler vector bundle over the compactcomplex manifold M with √− ∂ ¯ ∂ log G ≥ . If the horizontal (1 , -form T satisfies the nulleigenvector assumption, then √− ∂ ¯ ∂ log G ( t ) ≥ along the flow (0.1) for all t ≥ such thatthe solution exists. In this paper, we will give two applications of Theorem 0.1.For the first application, we consider the case of curve, i.e. dim M = 1. In this case, anyHermitian metric on M is K¨ahler automatically, and one can prove that the (1 , T satisfies the null eigenvector assumption. By Theorem 0.1, we obtain Proposition 0.2. If M is a curve, then the semipositivity of the curvature of O P ( E ∗ ) (1) ispreserved along the flow (0.1). In particular, if G = h i ¯ j v i ¯ v j comes from a Hermitian metric ( h i ¯ j ) of E ∗ andΩ = p ∗ ω + ω F S (0.3)for a fixed Hermitian metric ω , by Remark 2.1 (1), (0.1) is equivalent to the following Hermitian-Yang-Mills flow: h − · ∂h∂t + Λ R h + ( r − I = 0( h i ¯ j ( t )) > ,h i ¯ j (0) = ( h ) i ¯ j . (0.4)By Proposition 0.2 and Definition 1.7, we have Corollary 0.3. If M is a curve, then the Griffiths semipositivity is preserved along theHermitian-Yang-Mills flow (0.4). For the second application, we assume that E = T M and take ω ( G ) = √− g α ¯ β dz α ∧ d ¯ z β , where ( g α ¯ β ) denotes the inverse of the matrix (cid:16) ∂ G∂v α ∂ ¯ v β (cid:17) . Let G = g α ¯ β v α ¯ v β be the stronglypseudoconvex complex Finsler metric on T ∗ M induced by the following K¨ahler metric ω = √− g ) α ¯ β dz α ∧ d ¯ z β . XUEYUAN WAN
Then the flow (0.1) is equivalent to the following K¨ahler-Ricci flow ∂ω∂t + Ric( ω ) + ( n − ω = 0 ,ω > ,ω (0) = ω . (0.5)The solution of (0.1) is induced from the K¨ahler metric ω = √− g α ¯ β dz α ∧ d ¯ z β . In this case,the (1 , T can also be proved satisfying the null eigenvector assumption (see [3, Page254, Claim 2.2]). From Theorem 0.1 and Definition 1.7, we can reprove the following Mok’sproposition, which is contained in [24, Proposition 1.1] (see also [3, Theorem 5.2.10]). Proposition 0.4 ([24, Proposition 1.1]) . If ( M, ω ) is a compact K¨ahler manifold with nonneg-ative holomorphic bisectional curvature, then the nonnegativity is preserved along the K¨ahler-Ricci flow (0.5). Remark 0.5.
By (0.4) and (0.5), the flow (0.1) is a natural generalization of both Hermitian-Yang-Mills flow and K¨ahler-Ricci flow. And there are some other flows, which are also thegeneralization of the K¨ahler-Ricci flow. For example, Gill [14] introduced the
Chern-Ricci flow on Hermitian manifolds, and many properties of the flow were established in [30, 31]. Especially,Yang [35] proved the nonnegativity of the holomorphic bisectional curvature is not necessarilypreserved under the Chern-Ricci flow. In [29], Streets and Tian introduced the
Hermitiancurvature flow , proved short time existence for this flow, and derive basic long time blowupand regularity results. For a particular version of the Hermitian curvature flow, Ustinovskiy[32] proved the the property of Griffiths positive (nonnegative) Chern curvature is preservedalong this flow.This article is organized as follows. In Section 1, we shall fix the notation and recall somebasic definitions and facts on complex Finsler vector bundles, Griffiths positive (semipositive),and maximal principle for real (1 , P ( E ∗ ) and study the positivity along this flow, Theorem 0.1 would be proved inthis section. In Section 3, we will give two applications of Theorem 0.1, and we will proveProposition 0.2, Corollary 0.3 and Proposition 0.4. Acknowledgements.
The author would like to thank Professor Kefeng Liu and ProfessorHuitao Feng for their guidance over the years, and thank Professor Xiaokui Yang for manyvaluable discussions. 1.
Preliminaries
In this section, we shall fix the notation and recall some basic definitions and facts oncomplex Finsler vector bundles, Griffiths positive (semipositive), and maximal principle forreal (1 , Complex Finsler vector bundle.
Let M be a compact complex manifold of dimension n , and let π : E → M be a holomorphic vector bundle of rank r over M . Let z = ( z , · · · , z n ) OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 5 be a local coordinate system in M , and { e i } ≤ i ≤ r be a local holomorphic frame of E . Withrespect to the local frame of E , an element of E can be written as v = v i e i ∈ E, where we adopt the summation convention of Einstein. In this way, one gets a local coordinatesystem of the complex manifold E :( z ; v ) = ( z , · · · , z n ; v , · · · , v r ) . (1.1) Definition 1.1 ([19]) . A Finsler metric G on the holomorphic vector bundle E is a continuousfunction G : E → R satisfying the following conditions: F1): G is smooth on E o = E \ O , where O denotes the zero section of E ; F2): G ( z, v ) ≥ z, v ) ∈ E with z ∈ M and v ∈ π − ( z ), and G ( z, v ) = 0 if andonly if v = 0; F3): G ( z, λv ) = | λ | G ( z, v ) for all λ ∈ C .Moreover, G is called strongly pseudoconvex if F4): the Levi form √− ∂ ¯ ∂G on E o is positive-definite along each fiber E z = π − ( z ) for z ∈ M .Clearly, any Hermitian metric on E is naturally a strongly pseudoconvex complex Finslermetric on it.We write G i = ∂G/∂v i , G ¯ j = ∂G/∂ ¯ v j , G i ¯ j = ∂ G/∂v i ∂ ¯ v j ,G iα = ∂ G/∂v i ∂z α , G i ¯ j ¯ β = ∂ G/∂v i ∂ ¯ v j ∂ ¯ z β , etc., to denote the differentiation with respect to v i , ¯ v j (1 ≤ i, j ≤ r ), z α , ¯ z β (1 ≤ α, β ≤ n ). In thefollowing lemma we collect some useful identities related to a Finsler metric G . Lemma 1.2 ([10, 19]) . The following identities hold for any ( z, v ) ∈ E o , λ ∈ C : G i ( z, λv ) = ¯ λG i ( z, v ) , G i ¯ j ( z, λv ) = G i ¯ j ( z, v ) = G j ¯ i ( z, v ); G i ( z, v ) v i = G ¯ j ( z, v )¯ v j = G i ¯ j ( z, v ) v i ¯ v j = G ( z, v ); G ij ( z, v ) v i = G i ¯ jk ( z, v ) v i = G i ¯ j ¯ k ( z, v )¯ v j = 0 . If G is a strongly pseudoconvex complex Finsler metric on M , then there is a canonical h-vdecomposition of the holomorphic tangent bundle T E o of E o (see [10, §
5] or [12, § T E o = H ⊕ V . In terms of local coordinates, H = span C (cid:26) δδz α = ∂∂z α − G α ¯ j G ¯ jk ∂∂v k , ≤ α ≤ n (cid:27) , V = span C (cid:26) ∂∂v i , ≤ i ≤ r (cid:27) . The dual bundle T ∗ E o also has a smooth h-v decomposition T ∗ E o = H ∗ ⊕ V ∗ : H ∗ = span C { dz α , ≤ α ≤ n } , V ∗ = span C { δv i = dv i + G ¯ ji G α ¯ j dz α , ≤ i ≤ r } . (1.2) XUEYUAN WAN
Moreover, the differential operators ∂ V = ∂∂v i ⊗ δv i , ∂ H = δδz α ⊗ dz α . (1.3)are well-defined.With respect to the h-v decomposition (1.2), the (1 , √− ∂ ¯ ∂ log G has the follow-ing decomposition. For readers’ convenience, we give a proof of the following lemma due toKobayashi and Aikou (cf. [19, 4]). Lemma 1.3 ([19, 4]) . Let G be a strongly pseudoconvex complex Finsler metric on E . Onehas √− ∂ ¯ ∂ log G = − Ψ + ω V , where Ψ is called the Kobayashi curvature (see [12, Definition 1.2] ), Ψ = √− R i ¯ jα ¯ β v i ¯ v j G dz α ∧ d ¯ z β , ω V = √− ∂ log G∂v i ∂ ¯ v j δv i ∧ δ ¯ v j , (1.4) with R i ¯ jα ¯ β = − ∂ G i ¯ j ∂z α ∂ ¯ z β + G ¯ lk ∂G i ¯ l ∂z α ∂G k ¯ j ∂ ¯ z β . Proof.
By (1.2), we have ∂ log G∂v i ∂ ¯ v j δv i ∧ δ ¯ v j = ∂ log G∂v i ∂ ¯ v j ( dv i + G α ¯ l G ¯ li dz α ) ∧ ( d ¯ v j + G ¯ βk G ¯ jk d ¯ z β )= ∂ log G∂v i ∂ ¯ v j dv i ∧ d ¯ v j + ∂ log G∂v i ∂ ¯ v j G ¯ βk G ¯ jk dv i ∧ d ¯ z β + ∂ log G∂v i ∂ ¯ v j G α ¯ l G ¯ li dz α ∧ d ¯ v j + ∂ log G∂v i ∂ ¯ v j G α ¯ l G ¯ li G k ¯ β G ¯ jk dz α ∧ d ¯ z β . (1.5)For the last three terms in the RHS of (1.5), one has from Lemma 1.2 ∂ log G∂v i ∂ ¯ v j G ¯ βk G ¯ jk dv i ∧ d ¯ z β = GG i ¯ j − G i G ¯ j G G ¯ βk G ¯ jk dv i ∧ d ¯ z β = ∂ log G∂v i ∂ ¯ z β dv i ∧ d ¯ z β , (1.6) ∂ log G∂v i ∂ ¯ v j G α ¯ l G ¯ li dz α ∧ d ¯ v j = GG i ¯ j − G i G ¯ j G G α ¯ l G ¯ li dz α ∧ d ¯ v j = ∂ log G∂z α ∂ ¯ v j dz α ∧ d ¯ v j (1.7)and ∂ log G∂v i ∂ ¯ v j G α ¯ l G ¯ li G k ¯ β G ¯ jk dz α ∧ d ¯ z β = GG i ¯ j − G i G ¯ j G G α ¯ l G ¯ li G k ¯ β G ¯ jk dz α ∧ d ¯ z β = 1 G ( GG ¯ lk G α ¯ l G k ¯ β − G α G ¯ β ) dz α ∧ d ¯ z β . (1.8) OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 7
Submitting (1.6), (1.7) and (1.8) into (1.5), we obtain ∂ log G∂v i ∂ ¯ v j δv i ∧ δ ¯ v j = ∂ log G∂v i ∂ ¯ v j dv i ∧ d ¯ v j + ∂ log G∂v i ∂ ¯ z β dv i ∧ d ¯ z β + ∂ log G∂z α ∂ ¯ v j dz α ∧ d ¯ v j + 1 G ( GG ¯ lk G α ¯ l G k ¯ β − G α G ¯ β ) dz α ∧ d ¯ z β = ∂ ¯ ∂ log G + 1 G ( G ¯ lk G α ¯ l G k ¯ β − G α ¯ β ) dz α ∧ d ¯ z β = ∂ ¯ ∂ log G + ( G ¯ lk G αi ¯ l G k ¯ j ¯ β − G i ¯ jα ¯ β ) v i ¯ v j G dz α ∧ d ¯ z β = ∂ ¯ ∂ log G − √− , which completes the proof. (cid:3) Let q denote the natural projection q : E o → P ( E ) := E o / C ∗ ( z ; v ) ( z ; [ v ]) := ( z , · · · , z n ; [ v , · · · , v n ]) , which gives a local coordinate system of P ( E ) by( z ; w ) := ( z , · · · , z n ; w , · · · , w r − ) = (cid:18) z , · · · , z n ; v v k , · · · , v k − v k , v k +1 v k , · · · , v n v k (cid:19) (1.9)on U k := { ( z, [ v ]) ∈ P ( E ) , v k = 0 } .Denote by ((log G ) ab ) ≤ a,b ≤ r − the inverse of the matrix (cid:16) (log G ) a ¯ b := ∂ log G∂w a ¯ w b (cid:17) ≤ a,b ≤ r − andset δw a = dw a + (log G ) α ¯ b (log G ) ¯ ba dz α . One can define a vertical (1 , P ( E ) by ω F S := √− ∂ log G∂w a ∂ ¯ w b δw a ∧ δ ¯ w b , (1.10)which is well-defined (see e.g. [13, Section 1]). Moreover, Lemma 1.4.
By the pullback q ∗ : A , ( P ( E )) → A , ( E o ) , one has q ∗ ω F S = ω V . (1.11) Proof.
We only need to prove (1.11) at one point ( z, [ v ]) ∈ U k . Without loss of generality, weassume that k = r . For any point ( z, [ v ]) ∈ U r , one has from (1.9) q ∗ (cid:18) ∂∂v r (cid:19) = − r − X a =1 v a ( v r ) ∂∂w a , q ∗ (cid:18) ∂∂v b (cid:19) = 1 v r ∂∂w b , ≤ b ≤ r − . (1.12) XUEYUAN WAN
By (1.12), one has ∂ log G∂w a ∂ ¯ w b = ( ∂ ¯ ∂ log G )( ∂∂w a , ∂∂ ¯ w b )= ( ∂ ¯ ∂ log G )( q ∗ ( v r ∂∂v a ) , q ∗ (¯ v r ∂∂ ¯ v b ))= q ∗ ( ∂ ¯ ∂ log G )( v r ∂∂v a , ¯ v r ∂∂ ¯ v b )= | v r | ∂ log G∂v a ∂ ¯ v b . (1.13)Similarly, ∂ log G∂z α ∂ ¯ w b = ¯ v r ∂ log G∂z α ∂ ¯ v b , ∂ log G∂w a ∂ ¯ z β = v r ∂ log G∂v a ∂ ¯ z β (1.14)and ∂ log G∂v a ∂ ¯ v r = − | v r | ¯ v b ¯ v r ∂ log G∂w a ∂ ¯ w b , ∂ log G∂v r ∂ ¯ v b = − | v r | v a v r ∂ log G∂w a ∂ ¯ w b , ∂ log G∂v r ∂ ¯ v r = v a ¯ v b | v r | ∂ log G∂w a ∂ ¯ w b . (1.15)By a direct checking, one has(log G ) ¯ ba = G | v r | (cid:18) − v a v r G ¯ br + G ¯ ba + ¯ v b v a | v r | G ¯ rr − ¯ v b ¯ v r G ¯ ra (cid:19) . (1.16)By (1.14) and (1.16), we have(log G ) a ¯ β (log G ) ¯ ba = (cid:18) v r G ¯ bi − ¯ v b (¯ v r ) G ¯ ri (cid:19) ( G i ¯ β − G G ¯ β G i ) = 1¯ v r G ¯ bi G i ¯ β − ¯ v b (¯ v r ) G ¯ ri G i ¯ β . So q ∗ ( δ ¯ w b ) = q ∗ ( d ¯ w b + (log G ) a ¯ β (log G ) ¯ ba d ¯ z β )= 1¯ v r ( d ¯ v b + G ¯ bi G i ¯ β d ¯ z β ) − ¯ v b (¯ v r ) ( d ¯ v r + G ¯ ri G i ¯ β d ¯ z β )= 1¯ v r δ ¯ v b − ¯ v b (¯ v r ) δ ¯ v r . (1.17)From (1.19), (1.15) and (1.17), we obtain q ∗ ω F S = q ∗ (cid:18) √− ∂ log G∂w a ∂ ¯ w b δw a ∧ δ ¯ w b (cid:19) = √− | v r | ∂ log G∂v a ∂ ¯ v b (cid:18) v r δv a − v a ( v r ) δv r (cid:19) (cid:18) v r δ ¯ v b − ¯ v b (¯ v r ) δ ¯ v r (cid:19) = √− ∂ log G∂v a ∂ ¯ v b (cid:18) δv a ∧ δ ¯ v b − ¯ v b ¯ v r δv a ∧ δ ¯ v r − v a v r δv r ∧ δ ¯ v b + v a ¯ v b | ¯ v r | δv r ∧ δ ¯ v r (cid:19) = √− r X i,j =1 ∂ log G∂v i ∂ ¯ v j δv i ∧ δ ¯ v j = ω V . (cid:3) OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 9
Remark 1.5. (1) By (1.12) and (1.16), we get q ∗ ( δδz α ) = q ∗ ( ∂∂z α − G α ¯ l G ¯ lk ∂∂v k )= ∂∂z α − ( 1 v r G α ¯ l − v a ( v r ) G α ¯ l G ¯ lr ) ∂∂w a = ∂∂z α − (log G ) α ¯ b (log G ) ¯ ba ∂∂w a . For convenience, we will identify δδz α with q ∗ ( δδz α ), and denote N aα := (log G ) α ¯ b (log G ) ¯ ba ,so δδz α = ∂∂z α − N aα ∂∂w a . (1.18)(2) For any smooth function f ∈ C ∞ ( P ( E )), the vertical Laplacian is defined by∆ V f := (log G ) ¯ ba ∂ ∂w a ∂ ¯ w b f. By identifying f with q ∗ f , one has from (1.12) and (1.16)∆ V f = (log G ) ¯ ba ∂ ∂w a ∂ ¯ w b f = GG i ¯ j ∂ ∂v i ∂ ¯ v j f. (3) Noticing that √− π ∂ ¯ ∂ log G is a (1 , P ( E ), which represents the first Chernclass c ( O P ( E ) (1)). And Ψ is also a (1 , P ( E ), combining Lemma 1.3 withLemma 1.4, one has when restricting to P ( E ) √− ∂ ¯ ∂ log G = − Ψ + ω F S . (1.19) Proposition 1.6.
A Finsler metric G is strongly pseudoconvex if and only if ω F S is positivealong each fiber of p : P ( E ) → M .Proof. By Definition 1.1, G is strongly pseudoconvex if ( G i ¯ j ) is a positive definite matrix, whichgives a inner product h· , ·i on the vertical subbundle V .Denote T = v i ∂∂v i . If G is strongly pseudoconvex, for any X = X i ∂∂v i , then( −√− ω V ( X, ¯ X ) = 1 G ( GG i ¯ j − G i G ¯ j ) X i ¯ X j = 1 G ( k X k k T k − |h X, T i| ) ≥ , the equality holds if and only if X = λT for some constant λ ∈ C . So ω V has r − ω V ( T, T ) = 0 and q ∗ ( T ) = 0, by Lemma 1.4, ω F S ispositive along each fiber of p : P ( E ) → M .Conversely, if ω F S is positive along each fiber, then ω V = q ∗ ω F S has r − ω V ( T, T ) = 0. So G i ¯ j X i ¯ X j = 1 G | G i X i | + G ( −√− ω V ( X, X ) ≥ . Moreover, G i ¯ j X i ¯ X j = 0 if and only if X = λT and G i X i = 0, if and only if λ = 0. So ( G i ¯ j ) isa positive definite matrix. (cid:3) Let ( h i ¯ j ) be a Hermitian metric on E with respect to a local holomorphic frame { e i } ≤ i ≤ r . Definition 1.7 ([15]) . The Chern curvature of the metric ( h i ¯ j ) is called Griffiths positive (resp.
Griffiths semipositive ) if R i ¯ jα ¯ β X i X j Y α Y β > ≥ X = X i e i ∈ E and Y = Y α ∂∂z α ∈ T M . Here R i ¯ jα ¯ β = − ∂ h i ¯ j ∂z α ∂ ¯ z β + h ¯ lk ∂h i ¯ l ∂z α ∂h k ¯ j ∂ ¯ z β denotes the Chern curvature of ( h i ¯ j ).For the case of E = T M , the Hermitian metric ( h i ¯ j ) is called has positive (nonnegative)holomorphic bisectional curvature if its Chern curvature is Griffiths positive (semipositive).The Hermitian metric ( h i ¯ j ) induces a strongly pseudoconvex complex Finsler metric on E ∗ by G := h i ¯ j v i ¯ v j . By Remark 1.5, we have the following decomposition √− ∂ ¯ ∂ log G = − Ψ + ω F S , where − Ψ = − R i ¯ jα ¯ β v i ¯ v j G √− dz α ∧ d ¯ z β = R k ¯ lα ¯ β G ¯ li v i G ¯ jk ¯ v j G √− dz α ∧ d ¯ z β . From Proposition 1.6, ω F S is positive along fibers. So
Proposition 1.8.
The Chern curvature of ( h i ¯ j ) is Griffiths positive (resp. semipositive)if andonly if √− ∂ ¯ ∂ log G is a positive (resp. semipositive) (1 , -form on P ( E ∗ ) . Maximum principle for real (1 , -forms. In this subsection, we recall the maximumprinciple for real (1 , Theorem 1.9 ([9, Theorem 4.6]) . Let ω ( t ) = √− g α ¯ β ( t ) dz α ∧ d ¯ z β be a smooth -parameterfamily of Hermitian metrics on a compact complex manifold M . Let η ( t ) = √− η α ¯ β ( t ) dz α ∧ d ¯ z β be a real (1 , -form satisfying ∂∂t η ≥ ∆ ω ( t ) η + σ, where ∆ ω ( t ) := g α ¯ β ∇ ∂∂zα ∇ ∂∂ ¯ zβ , ∇ denotes the Chern connection of ω ( t ) , σ ( α, ω, t ) is a real (1 , -form which is locally Lipschitz in all its arguments and satisfies the null eigenvectorassumption that ( −√− σ ( V, V )( z, t ) = ( σ α ¯ β V α V β )( z, t ) ≥ whenever V ( z, t ) = V α ∂∂z α is a null eigenvector of η ( t ) , that is whenever ( η α ¯ β V α )( z, t ) = 0 . If η (0) ≥ , then η ( t ) ≥ for all t ≥ such that the solution exists. OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 11
Proof.
Suppose that η > ≤ t < t , and ( z , t ) is a point and time and v = v α ∂∂z α isa vector such that η α ¯ β v α ( z , t ) = 0 . (1.20)Then η α ¯ β W α W β ( z, t ) ≥ z ∈ M t ∈ [0 , t ], and tangent vectors W ∈ T z M . By paralleltranslation, one can extend v to a vector field V defined in a neighborhood of ( z , t ) such that V ( z , t ) = v and ∂V∂t ( z , t ) = 0 , ∇ V ( z , t ) = 0 . (1.21)Then at the point ( z , t ), one has ∂∂t ( η α ¯ β V α V β ) = ( ∂∂t η α ¯ β ) V α V β ≥ (∆ ω ( t ) η α ¯ β + σ α ¯ β ) V α V β . Since η α ¯ β V α V β ( z , t ) = 0 and η α ¯ β V α V β ( z, t ) = 0 in the neighborhood of z , one has∆ ω ( t ) ( η α ¯ β V α V β ) ≥ . By (1.20) and (1.21), one has at the point ( z , t )∆ ω ( t ) ( η α ¯ β V α V β ) = (∆ ω ( t ) η α ¯ β ) V α V β + g γ ¯ δ ( ∇ ∂∂ ¯ zδ η α ¯ β )( ∇ ∂∂zγ V α ) V β + g γ ¯ δ ∇ ∂∂zγ ( η α ¯ β V α ) ∇ ∂∂ ¯ zδ V β + η α ¯ β V α ∆ ω ( t ) V β = (∆ ω ( t ) η α ¯ β ) V α V β . Combining with the null eigenvector assumption σ α ¯ β V α V β ( z , t ) ≥ , it follows that ∂∂t ( η α ¯ β V α V β ) ≥ ∆ ω ( t ) ( η α ¯ β V α V β ) + σ α ¯ β V α V β ≥ z , t ). Hence, if η α ¯ β V α V β ever becomes zero, it cannot decrease further. So η ( t ) ≥ t ≥ (cid:3) A flow over projective bundle
Definition of the flow.
Let M be a compact complex manifold of dimension n , let π : E → M be a holomorphic vector bundle of rank r over M . Let E ∗ denote the dual bundleof E , { e i } ri =1 be a local holomorphic frame of E ∗ , then( z, v ) = ( z , · · · , z n ; v , · · · , v r )gives a local coordinate system of the complex manifold E ∗ , which represents the point v i e i ∈ E ∗ . For any strongly pseudoconvex complex Finsler metric G on E ∗ , by Remark 1.5, we havethe following decomposition √− ∂ ¯ ∂ log G = − Ψ + ω F S . By Proposition 1.6, ω F S is positive along each fiber of p : P ( E ∗ ) → M . Let ω ( G ) = √− g α ¯ β ( G ) dz α ∧ d ¯ z β be a horizontal (1 , P ( E ∗ ) depending smoothly on the Finsler metric G , which ispositive on horizontal directions, namely ( g α ¯ β ( G )) is a positive definite matrix.Then one can define a Hermitian metric on P ( E ∗ ) byΩ := ω ( G ) + ω F S . Let G be a strongly pseudoconvex complex Finsler metric on E ∗ , we consider the followingflow: ∂∂t log G = ∆ Ω log G,ω
F S > ,G (0) = G . (2.1)Here ∆ Ω := √− ∂ ¯ ∂ , Λ is the adjoint operator of Ω ∧ • .Since ∆ Ω log G = Λ √− ∂ ¯ ∂ log G = Λ ω ( G ) ( − Ψ) + ( r − P ( E ∗ ), so along the flow G ( t ) = e R t ∆ Ω log Gdt G , ω F S > . By Definition 1.1, one see that G ( t ) is always a strongly pseudoconvex complex Finsler metricon E ∗ . Remark 2.1. (1) For the case ω ( G ) = √− g α ¯ β dz α ∧ d ¯ z β is a fixed K¨ahler metric on M , and G = h i ¯ j v i ¯ v j comes from a Hermitian metric ( h i ¯ j )of E ∗ , then 0 = ∂∂t log G − ∆ Ω log G = 1 G ∂G∂t + Λ ω Ψ − ( r − v i v j G ∂h i ¯ j ( t ) ∂t + g α ¯ β R i ¯ jα ¯ β − ( r − h ¯ ji ( t ) ! , (2.3) where h i ¯ j ( t ) := ∂ G ( t ) ∂v i ∂ ¯ v j . From above equation, one has ∂h i ¯ j ( t ) ∂t + g α ¯ β ∂ ∂v i ∂ ¯ v j ( R i ¯ jα ¯ β v i ¯ v j ) − ( r − h ¯ ji ( t ) = 0 . Since h i ¯ j (0) = h i ¯ j is a Hermitian metric, which is independent of the vertical coordinates { v i , ≤ i ≤ r } , so h i ¯ j ( t ) is also a Hermitian metric. In fact, by induction, we assume OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 13 that ( ∂ k h i ¯ j ∂t k ) | t =0 is independent of fibers, then ∂ k +1 h i ¯ j ( t ) ∂t k +1 | t =0 = ∂ k ∂t k (cid:18) − g α ¯ β ∂ ∂v i ∂ ¯ v j ( R i ¯ jα ¯ β v i ¯ v j ) + ( r − h ¯ ji ( t ) (cid:19) | t =0 = − g α ¯ β ∂ ∂v i ∂ ¯ v j (cid:18) ( ∂ k ∂t k R i ¯ jα ¯ β ) | t =0 v i ¯ v j (cid:19) + ( r − ∂ k ∂t k h ¯ ji ) | t =0 = − g α ¯ β ( ∂ k ∂t k R i ¯ jα ¯ β ) | t =0 + ( r − ∂ k ∂t k h ¯ ji ) | t =0 , which is also independent of fibers, because ∂ k ∂t k R i ¯ jα ¯ β is the combination by h ¯ ji and ∂ l ∂t l h ¯ ji , ≤ l ≤ k . It follows that h ¯ ji ( t ) = h ¯ ji (0) + ∂h i ¯ j ( t ) ∂t | t =0 t + · · · + ∂ k h i ¯ j ( t ) ∂t k | t =0 t k + · · · is independent of fibers for small t . By Proposition 1.6 and ω F S > h ¯ ji ( t ) is a positivedefinite matrix, so ( h ¯ ji ( t )) is a Hermitian metric on E ∗ .Therefore, (2.3) is equivalent to ∂h i ¯ j ( t ) ∂t + g α ¯ β R i ¯ jα ¯ β − ( r − h ¯ ji ( t ) = 0 . (2.4) Multiplying by h k ¯ j to both sides of (2.4), one has h − · ∂h∂t + Λ R + ( r − I := h ¯ ji ∂h k ¯ j ∂t + g α ¯ β R ikα ¯ β + ( r − δ ik = 0 , which is exactly the Hermitian-Yang-Mills flow [1, 11] (see also [20]).(2) For the case of E = T M . Let ω ( G ) = √− g α ¯ β dz α ∧ d ¯ z β , where ( g α ¯ β ) denotes the inverse of the matrix (cid:16) ∂ G∂v α ∂ ¯ v β (cid:17) . Let G = g α ¯ β v α ¯ v β be a stronglypseudoconvex complex Finsler metric on T ∗ M induced by the Hermitian metric ω = √− g ) α ¯ β dz α ∧ d ¯ z β . Similar to the above case, the flow (2.1) is equivalent to ∂g γ ¯ δ ∂t + g α ¯ β R γ ¯ δα ¯ β + ( r − g γ ¯ δ = 0 , (2.5) which is exactly the equation given by [22, (7.11)]. By [22, Theorem 7.1] (or [22, Remark7.2]), if the initial metric ω is a K¨ahler metric, then this flow is reduced to the usualK¨hler-Ricci flow (see [6]). Positivity preserving along the flow.
In this subsection, we shall discuss the posi-tivity preserving along the flow (2.1). We assume that the initial metric G (0) = G satisfies √− ∂ ¯ ∂ log G ≥ . Let ǫ > ǫ = ω ( G ) + ǫ √− ∂ ¯ ∂ log G = √− g α ¯ β − ǫ Ψ α ¯ β ) dz α ∧ d ¯ z β + ǫ √− ∂ log G∂w a ∂ ¯ w b δw a ∧ δ ¯ w b (2.6)is a Hermitian metric on P ( E ∗ ), where Ψ α ¯ β is given by Ψ = √− α ¯ β dz α ∧ d ¯ z β . Denote by ∇ ǫ the Chern connection of the Hermitian metric Ω ǫ , which is the unique connection preservingthe holomorphic structure and the metric Ω ǫ . For any two (1 , X, Y of P ( E ∗ ), then ∇ ǫX Y = h∇ ǫX Y, δδz β i ǫ g ¯ βαǫ δδz α + h∇ ǫX Y, ∂∂w b i ǫ ǫ (log G ) ¯ ba ∂∂w a = X h Y, δδz β i ǫ g ¯ βαǫ δδz α − h Y, X ( δδz β ) i ǫ g ¯ βαǫ δδz α + X h Y V , ∂∂w b i ǫ ǫ (log G ) ¯ ba ∂∂w a = X h Y, δδz β i ǫ g ¯ βαǫ δδz α + h Y V , X ( N bβ ) ∂∂w b i ǫ g ¯ βαǫ δδz α + X h Y V , ∂∂w b i ǫ ǫ (log G ) ¯ ba ∂∂w a , (2.7)where Y V denotes the vertical part of Y , ( g ¯ βα ) ǫ denotes the inverse of g ǫα ¯ β := g α ¯ β − ǫ Ψ α ¯ β , h· , ·i ǫ is the inner product defined by Ω ǫ .Denote by Z A the coordinates z α or w a , 1 ≤ A ≤ n + r −
1. We rewrite Ω ǫ as the followingform: Ω ǫ = √− ǫA ¯ B dZ A ∧ d ¯ Z B . (2.8)In this form, the Chern connection is given by ∇ ǫ ∂∂ZA ∂∂Z B = Γ CAB ∂∂Z C , Γ CAB = ∂ Ω ǫB ¯ D ∂Z A Ω ¯ DCǫ . (2.9)Here (Ω ¯ DCǫ ) denotes the inverse of the matrix (Ω ǫC ¯ D ). The Chern curvature tensor of Ω ǫ isdefined by R ǫA ¯ BC ¯ D := R ( ∂∂Z A , ∂∂ ¯ Z B , ∂∂Z C , ∂∂ ¯ Z D ):= h ∂∂Z A , ( ∇ ǫ ∂∂ZD ∇ ǫ ∂∂ ¯ ZC − ∇ ǫ ∂∂ ¯ ZC ∇ ǫ ∂∂ZD − ∇ ǫ [ ∂∂ZD , ∂∂ ¯ ZC ] ) ∂∂Z B i ǫ = − Ω ǫA ¯ E ∂ Γ EDB ∂Z C . (2.10)The Chern connection ∇ ǫ induces a natural connection on the cotangent bundle T ∗ P ( E ∗ ) (resp. T ∗ P ( E ∗ ) ) by ∇ ǫA ( f C dZ C ) = ( ∂ A f C − Γ BAC f B ) dZ C , (resp. ∇ ǫ ¯ B ( f ¯ D d ¯ Z D ) = ( ∂ ¯ B f ¯ D − Γ EBD f ¯ E ) dZ C )(2.11)for any smooth (1 , f C dZ C (resp. (0 , f ¯ D d ¯ Z D ), where ∇ ǫA := ∇ ǫ ∂∂ZA . For conve-nience, we denote ∇ ǫA f C := ∂ A f C − Γ BAC f B , ∇ ǫ ¯ B f ¯ D := ∂ ¯ B f ¯ D − Γ EBD f ¯ E (2.12) OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 15 and ( ∇ ǫA f C ¯ D ) dZ C ∧ d ¯ Z D := ∇ ǫ ∂∂ZA ( f C ¯ D dZ C ∧ d ¯ Z D ) , ∇ ǫA f C ¯ D = ∂ A f C ¯ D − Γ BAC f B ¯ D . (2.13)By above notations, we have ∇ ǫA Ω ǫC ¯ D = ∂ A Ω ǫC ¯ D − Γ BAC Ω ǫB ¯ D = 0 . (2.14)By taking ∂ ¯ ∂ to the both sides of the first equation of (2.1), one has ∂∂t ∂ ¯ ∂ log G = ∂ ¯ ∂ ∆ Ω log G = ∂ ¯ ∂ ( tr ω ( G ) ( − Ψ)) , (2.15)where the last equality follows from (2.2). Since lim ǫ → ( ω ( G ) − ǫ Ψ) = ω ( G ), so ∂∂t ∂ ¯ ∂ log G = lim ǫ → ∂ ¯ ∂ ( tr ω ( G ) − ǫ Ψ ( − Ψ))= lim ǫ → ∂ ¯ ∂ ∆ Ω ǫ log G = lim ǫ → ∂ C ∂ ¯ D (Ω A ¯ Bǫ ∂ A ∂ ¯ B log G ) dZ C ∧ d ¯ Z D . (2.16)Denote f := log G and f A ¯ B := ∂ A ∂ ¯ B f, f A ¯ BC := ∂ A ∂ ¯ B ∂ C f, f A ¯ BC ¯ D := ∂ A ∂ ¯ B ∂ C ∂ ¯ D f, etc.. By (2.13) and (2.14), one has ∂ C ∂ ¯ D (Ω A ¯ Bǫ ∂ A ∂ ¯ B log G ) = ∂ C ∂ ¯ D (Ω A ¯ Bǫ f A ¯ B ) = ∇ ǫC ∇ ǫ ¯ D (Ω ABǫ f A ¯ B )= Ω A ¯ Bǫ ∇ ǫC ∇ ǫ ¯ D f A ¯ B = Ω A ¯ Bǫ ∇ ǫC ( f A ¯ B ¯ D − Γ EDB f A ¯ E )= Ω A ¯ Bǫ ( f A ¯ BC ¯ D − Γ FCA f F ¯ B ¯ D − ∂ C Γ EDB f AE − Γ EDB ( f AC ¯ E − Γ FCA f F ¯ E )) . (2.17)Similarly, Ω A ¯ Bǫ ( ∇ ǫA ∇ ǫ ¯ B f C ¯ D )= Ω A ¯ Bǫ ( f A ¯ BC ¯ D − Γ FAC f F ¯ B ¯ D − ∂ A Γ EBD f CE − Γ EBD ( f AC ¯ E − Γ FAC f F ¯ E )) . (2.18)Combining (2.17) with (2.18), we obtain ∂ C ∂ ¯ D (Ω A ¯ Bǫ f A ¯ B ) − Ω A ¯ Bǫ ( ∇ ǫA ∇ ǫ ¯ B f C ¯ D )= Ω A ¯ Bǫ (Γ FAC − Γ FCA ) f F ¯ D ¯ B + Ω A ¯ Bǫ (Γ EBD − Γ EDB ) f AC ¯ E + Ω A ¯ Bǫ (Γ EDB Γ FCA − Γ EBD Γ FAC ) f F ¯ E + Ω A ¯ Bǫ ∂ A Γ EBD f C ¯ E − Ω A ¯ Bǫ ∂ C Γ EDB f A ¯ E = Ω A ¯ Bǫ (Γ FAC − Γ FCA ) ∇ ¯ D f F ¯ B + Ω A ¯ Bǫ (Γ EBD − Γ EDB ) ∇ A f C ¯ E − Ω A ¯ Bǫ Ω F ¯ Eǫ R F ¯ DA ¯ B f C ¯ E + Ω A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E . (2.19) Substituting (2.19) into (2.16), we have ∂∂t ∂ ¯ ∂ log G = lim ǫ → (cid:16) Ω A ¯ Bǫ ∇ ǫA ∇ ǫ ¯ B ( ∂ ¯ ∂ log G )+ (cid:16) Ω A ¯ Bǫ (Γ FAC − Γ FCA ) ∇ ¯ D f F ¯ B + Ω A ¯ Bǫ (Γ EBD − Γ EDB ) ∇ A f C ¯ E − Ω A ¯ Bǫ Ω F ¯ Eǫ R F ¯ DA ¯ B f C ¯ E + Ω A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E (cid:17) dZ C ∧ d ¯ Z D (cid:17) . (2.20)As in the proof of Theorem 1.9, we assume that √− ∂ ¯ ∂ log G ≥ ≤ t < t , and( z , [ v ] , t ) is a point and time and u = u A ∂∂Z A is a vector such that f C ¯ D u C ( z , [ v ] , t ) = ( ∂ C ∂ ¯ D log G ) u C ( z , [ v ] , t ) = 0(2.21)and ( ∂ ¯ ∂ log G ( W, W ))( z, [ v ] , t ) = ( ∂ C ∂ ¯ D log G ) W C W D ( z, [ v ] , t ) ≥ z, [ v ]) ∈ P ( E ∗ ), t ∈ [0 , t ], and tangent vectors W ∈ T ( z, [ v ]) P ( E ∗ ). This implies that u = u α δδz α ∈ q ∗ H . (2.23)Indeed, one may assume that u = u + u , where u = u α δδz α , u = u a ∂∂w a are the horizontaland vertical parts of u respectively. By (1.19) and ω F S >
0, one has0 = ( √− ∂ ¯ ∂ log G )( u, ¯ u )= ( √− ∂ ¯ ∂ log G )( u + u , u + u )= ( √− ∂ ¯ ∂ log G )( u , u ) + ω F S ( u , u ) ≥ ω F S ( u , u ) ≥ , (2.24)and all equalities hold if and only if u = 0, namely u = u . From Lemma 1.3 and (2.23),(2.21) is equivalent to ( i u Ψ)( z , [ v ] , t ) = 0 . (2.25)For any ǫ >
0, by parallel translation, one can extend u to a vector field U ǫ = U Aǫ ∂∂Z A definedin a neighborhood of ( z , [ v ] , t ) such that U ǫ ( z , [ v ] , t ) = u and ∂U ǫ ∂t ( z , [ v ] , t ) = 0 , ( ∇ ǫ U ǫ )( z , [ v ] , t ) = 0 . (2.26)This can be done by parallel translating u along radial rays with respect to the connection ∇ ǫ ,and then by extending to be independent of time t .We assume that U ǫ ( z, [ v ] , t ) = U αǫ δδz α + U aǫ ∂∂w a . By (2.7), one has ∇ ǫ U ǫ = ¯ ∂U ǫ + (cid:16) ∂ ( U αǫ g ǫα ¯ β ) + U aǫ ∂ ( N bβ ) ǫ (log G ) a ¯ b (cid:17) g ¯ βγǫ δδz γ + ∂ ( U aǫ (log G ) a ¯ b )(log G ) ¯ bc ∂∂w c . (2.27) OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 17
So the second equation of (2.26) is equivalent to ¯ ∂U αǫ = 0 , ¯ ∂ ( − U αǫ N aα + U aǫ ) = 0 ,∂ ( U αǫ g ǫα ¯ β ) + U aǫ ∂ ( N bβ ) ǫ (log G ) a ¯ b = 0 ,∂ ( U aǫ (log G ) a ¯ b )(log G ) ¯ bc = 0 , (2.28)at the point ( z , [ v ] , t ). By (2.23), U aǫ = 0 at the point ( z , [ v ] , t ), so (2.28) is equivalent to ¯ ∂U αǫ = 0 , ¯ ∂U aǫ = u α ¯ ∂N aα ,∂U αǫ + g ¯ βαǫ ∂g ǫγ ¯ β u γ = 0 ,∂U aǫ = 0 . (2.29)Since lim ǫ → g ǫα ¯ β = g α ¯ β , so lim ǫ → U ǫ = U which satisfies the following equations: ¯ ∂U α = 0 , ¯ ∂U a = u α ¯ ∂N aα ,∂U α + g ¯ βα ∂g γ ¯ β u γ = 0 ,∂U a = 0(2.30)and ∂U∂t = 0 at the point ( z , [ v ] , t ). By (2.20), (2.21) and (2.26), one has at the point( z , [ v ] , t ), ∂∂t (cid:0) ∂ ¯ ∂ log G ( U, U ) (cid:1) = lim ǫ → (cid:16) Ω A ¯ Bǫ ∇ ǫA ∇ ǫ ¯ B ( ∂ ¯ ∂ log G )( u, u )+ (cid:16) Ω A ¯ Bǫ (Γ FAC − Γ FCA ) ∇ ¯ D f F ¯ B + Ω A ¯ Bǫ (Γ EBD − Γ EDB ) ∇ A f C ¯ E − Ω A ¯ Bǫ Ω F ¯ Eǫ R F ¯ DA ¯ B f C ¯ E + Ω A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E (cid:17) u C ¯ u D (cid:17) = lim ǫ → (cid:16) Ω A ¯ Bǫ ∂ A ∂ ¯ B ( ∂ ¯ ∂ log G ( U ǫ , U ǫ ))+ (cid:16) Ω A ¯ Bǫ (Γ FAC − Γ FCA ) ∇ ¯ D f F ¯ B + Ω A ¯ Bǫ (Γ EBD − Γ EDB ) ∇ A f C ¯ E +Ω A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E (cid:17) u C ¯ u D (cid:17) . (2.31)In order to deal with (2.31), we assume that ω ( G ) = p ∗ ω for some K¨ahler metric ω on M ,so Ω ǫ = ω ( G ) + ǫ √− ∂ ¯ ∂ log G (2.32)is a K¨ahler metric on P ( E ∗ ) for ǫ > ∂∂t (cid:0) ∂ ¯ ∂ log G ( U, U ) (cid:1) = lim ǫ → (cid:16) Ω A ¯ Bǫ ∂ A ∂ ¯ B ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) + Ω A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E u C ¯ u D (cid:17) . (2.33) For the first term in the RHS of (2.33), we haveΩ A ¯ Bǫ ∂ A ∂ ¯ B ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) = √− ∂ ¯ ∂ ( ∂ ¯ ∂ log G ( U ǫ , U ǫ ))= ∆ H Ω ǫ ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) + ∆ V Ω ǫ ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) . (2.34)Here ∆ V Ω ǫ = ǫ (log G ) ¯ ba ∂ ∂w a ∂ ¯ w b is the vertical Laplacian (see Remark 1.5), while the hor-izontal Laplacian ∆ H Ω ǫ is defined by ∆ H Ω ǫ ϕ = g α ¯ βǫ ( ∂ ¯ ∂ϕ )( δδz α , δδ ¯ z β ) for any smooth function ϕ ∈ C ∞ ( P ( E ∗ )). Since lim ǫ → o g ǫα ¯ β = g α ¯ β and lim ǫ → U ǫ = U , solim ǫ → ∆ H Ω ǫ ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) = ∆ H Ω ( ∂ ¯ ∂ log G ( U, U )) . (2.35)By Remark 1.5 (1), one has∆ V Ω ǫ ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) = ∆ V Ω ǫ ( f c ¯ d U cǫ U dǫ ) + ∆ V Ω ǫ (( − Ψ) α ¯ β U αǫ U βǫ )= 1 ǫ f a ¯ b ∂ ∂w a ∂ ¯ w b ( f c ¯ d U cǫ U dǫ ) + 1 ǫ f a ¯ b ∂ ∂w a ∂ ¯ w b (( − Ψ) α ¯ β U αǫ U βǫ ) . (2.36)For the first term in the RHS of (2.36), by (2.29) and U c = 0 at the point ( z , [ v ] , t ), wehave 1 ǫ f a ¯ b ∂ ∂w a ∂ ¯ w b ( f c ¯ d U cǫ U dǫ ) = 1 ǫ f a ¯ b f c ¯ d ∂ ¯ b U cǫ ∂ a U dǫ = 1 ǫ f a ¯ b f c ¯ d u α u β ∂ ¯ b N cα ∂ a N dβ . (2.37)The following lemma is actually proved in [34, (3.46)]. For readers’ convenience, we give aproof here. Lemma 2.2. f ¯ ba ∂ ∂w a ∂ ¯ w b ( − Ψ) α ¯ β = ∂ ¯ ∂ log det( f a ¯ b )( δδz α , δδ ¯ z β ) − h ¯ ∂ V δδz α , ¯ ∂ V δδz β i , (2.38) where ¯ ∂ V δδz α := ∂∂ ¯ w d ( − f α ¯ b f ¯ bc ) δ ¯ w d ⊗ ∂∂w c and h ¯ ∂ V δδz α , ¯ ∂ V δδz β i := f ¯ ba ∂ ¯ b N cα ∂ a N dβ f c ¯ d .Proof. Let ( − Ψ) α ¯ β denote the coefficient of − Ψ, i.e. − Ψ = √− − Ψ) α ¯ β dz α ∧ d ¯ z β , then( − Ψ) α ¯ β = f α ¯ β − f α ¯ d f ¯ dc f c ¯ β . (2.39)In fact, by the decomposition (1.19), one has( − Ψ) α ¯ β = ( −√− − Ψ)( δδz α , δδ ¯ z β )= ( ∂ ¯ ∂f )( ∂∂z α − f α ¯ b f ¯ ba ∂∂w a , ∂∂ ¯ z β − f ¯ βa f a ¯ b ∂∂ ¯ w b )= f α ¯ β − f α ¯ d f ¯ dc f c ¯ β , which proves (2.40). OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 19
For any fixed point ( z, [ v ]) ∈ P ( E ∗ ) | z , z ∈ M , we take normal coordinates near ( z, [ v ]) suchthat f a ¯ b ( z, [ v ]) = δ ab , f a ¯ bc ( z, [ v ]) = 0. Evaluating at ( z, [ v ]) we see that f ¯ ba ∂ ∂w a ∂ ¯ w b ( − Ψ) α ¯ β = f ¯ ba ∂ ∂w a ∂ ¯ w b ( f α ¯ β − f α ¯ d f ¯ dc f c ¯ β )= f ¯ ba (cid:0) f a ¯ bα ¯ β − f α ¯ ca ¯ b f c ¯ β − f α ¯ c f c ¯ βa ¯ b + f α ¯ d f a ¯ bd ¯ c f c ¯ β − f α ¯ ca f c ¯ β ¯ b − f α ¯ c ¯ b f c ¯ βa (cid:1) = f ¯ ba ( − ¯ ∂ ( ∂f a ¯ d f ¯ dc ) f c ¯ b )( δδz α , δδ ¯ z β ) − f ¯ ba ∂ ¯ b ( − f α ¯ c f ¯ cd ) ∂ a ( − f k ¯ β f k ¯ l ) f d ¯ l = ∂ ¯ ∂ log det( f a ¯ b )( δδz α , δδ ¯ z β ) − h ¯ ∂ V δδz α , ¯ ∂ V δδz β i . which completes the proof. (cid:3) By Lemma 2.2 and (2.29), one has1 ǫ f a ¯ b ∂ ∂w a ∂ ¯ w b (( − Ψ) α ¯ β U αǫ U βǫ )= 1 ǫ f a ¯ b (cid:16) ∂ a ∂ ¯ b ( − Ψ) α ¯ β u α ¯ u β + ∂ a ( − Ψ) α ¯ β ∂ ¯ b U βǫ u α + ∂ ¯ b ( − Ψ) α ¯ β ∂ a U αǫ u β + ∂ a U αǫ ∂ b U bǫ ( − Ψ) α ¯ β (cid:17) = 1 ǫ ∂ ¯ ∂ log det( f a ¯ b )( u, ¯ u ) − ǫ f a ¯ b f c ¯ d u α u β ∂ ¯ b N cα ∂ a N dβ − f a ¯ b ∂ a Ψ α ¯ β ∂ b Ψ γ ¯ τ u α u γ g τ ¯ β + O ( ǫ ) , (2.40)where the last equality follows from ∂ a U αǫ = − g ¯ βαǫ ∂ a g ǫγ ¯ β u γ = ǫg ¯ βαǫ ∂ a Ψ γ ¯ β u γ = O ( ǫ ).Substituting (2.37) and (2.40) into (2.36), we obtain∆ V Ω ǫ ( ∂ ¯ ∂ log G ( U ǫ , U ǫ )) = 1 ǫ ∂ ¯ ∂ log det( f a ¯ b )( u, ¯ u ) − (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) + O ( ǫ ) . (2.41)Here we denote (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) := f a ¯ b ∂ a Ψ α ¯ β ∂ b Ψ γ ¯ τ u α u γ g τ ¯ β .For the second term in the RHS of (2.33), we haveΩ A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E u C ¯ u D = ( − Ψ) α ¯ δ g α ¯ βǫ g γ ¯ δǫ R ǫγ ¯ βσ ¯ τ u σ ¯ u τ + 1 ǫ f a ¯ b R ǫa ¯ bσ ¯ τ u σ ¯ u τ , (2.42)where R ǫγ ¯ βσ ¯ τ = R ǫ ( δδz γ , δδ ¯ z β , δδz σ δδ ¯ z τ ) and R ǫa ¯ bσ ¯ τ = R ǫ ( ∂∂w a , ∂∂ ¯ w b , δδz σ , δδ ¯ z τ ). By (2.7) and (2.10),one has R ǫγ ¯ βσ ¯ τ = R ǫ ( δδz γ , δδ ¯ z β , δδz σ , δδ ¯ z τ )= h δδz γ , ( ∇ ǫ δδzτ ∇ ǫ δδ ¯ zσ − ∇ ǫ δδ ¯ zσ ∇ ǫ δδzτ − ∇ ǫ [ δδzτ , δδ ¯ zσ ] ) δδz β i ǫ = (cid:16) ¯ ∂ ( ∂g ǫγ ¯ δ · g ¯ δαǫ ) g ǫα ¯ β (cid:17) ( δδz σ , δδ ¯ z τ ) − ǫf c ¯ d δδz σ N dβ δδ ¯ z τ N cγ = R gγ ¯ βσ ¯ τ + O ( ǫ ) , (2.43) where R gγ ¯ βσ ¯ τ := − ∂ g γ ¯ β ∂z σ ∂ ¯ z τ + g ¯ τ α ∂g α ¯ β ∂ ¯ z τ ∂g γ ¯ δ ∂z σ denotes the Chern curvature of the K¨ahler metric ω = √− g α ¯ β dz α ∧ d ¯ z β . And1 ǫ f a ¯ b R ǫa ¯ bσ ¯ τ u σ u τ = 1 ǫ f a ¯ b R ǫ ( ∂∂w a , ∂∂ ¯ w b , δδz σ , δδ ¯ z τ ) u σ u τ = 1 ǫ f a ¯ b h ∂∂w a , ( ∇ ǫ δδzτ ∇ ǫ δδ ¯ zσ − ∇ ǫ δδ ¯ zσ ∇ ǫ δδzτ − ∇ ǫ [ δδzτ , δδ ¯ zσ ] ) ∂∂w b i ǫ u σ u τ = 1 ǫ f a ¯ b (cid:16) ¯ ∂ ( ∂f a ¯ d · f ¯ dc ) f c ¯ b (cid:17) ( u, u ) + δδz σ N bβ δδ ¯ z τ N aα f a ¯ b g ¯ βαǫ u σ u τ = − ǫ ∂ ¯ ∂ log det( f a ¯ b )( u, u ) + (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) + O ( ǫ ) , (2.44)where the last equality follows from the following equality: δδ ¯ z γ N aα f a ¯ b = ( ∂ ¯ γ − f c ¯ γ f c ¯ e ∂ ¯ e )( f α ¯ d f ¯ da ) f a ¯ b = f a ¯ b ¯ γ − f α ¯ d f ¯ da f a ¯ b ¯ γ − f c ¯ γ f c ¯ e f ¯ eα ¯ b + f c ¯ γ f c ¯ e f α ¯ d f ¯ da f a ¯ b ¯ e = ∂ ¯ b ( f α ¯ γ − f α ¯ b f ¯ ba f a ¯ γ )= ∂ ¯ b ( − Ψ) α ¯ γ . Substituting (2.43) and (2.44) into (2.42), we haveΩ A ¯ Bǫ Ω F ¯ Eǫ R ǫF ¯ BC ¯ D f A ¯ E u C ¯ u D = − ǫ ∂ ¯ ∂ log det( f a ¯ b )( u, u ) + (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) + h R g ( u, u ) , − Ψ i Ω + O ( ǫ ) . (2.45)Here we denote h R g ( u, u ) , − Ψ i Ω = ( − Ψ) α ¯ δ g α ¯ β g γ ¯ δ R gγ ¯ βσ ¯ τ u σ ¯ u τ .Substituting (2.35), (2.41) and (2.45) into (2.33), we obtain ∂∂t (cid:0) ∂ ¯ ∂ log G ( U, U ) (cid:1) = ∆ H Ω ( ∂ ¯ ∂ log G ( U, U )) + h R g ( u, u ) , − Ψ i Ω − (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) . (2.46)at the point ( z , [ v ] , t ).Now we define a horizontal and real (1 , T as follow,( −√− T ( X, X ) := h R g ( X, X ) , − Ψ i Ω − (cid:12)(cid:12) i X ∂ V Ψ (cid:12)(cid:12) (2.47)for any horizontal vector X = X α δδz α . And we assume that T satisfies the null eigenvectorassumption (see Theorem 1.9), by Theorem 1.9, we obtain Theorem 2.3.
Let π : ( E ∗ , G ) → M be a holomorphic Finsler vector bundle over M with √− ∂ ¯ ∂ log G ≥ . Consider the following flow over the projective bundle p : P ( E ∗ ) → M : ∂∂t log G = ∆ Ω log G,ω
F S > ,G (0) = G , (2.48) where Ω = ω ( G ) + ω F S , ω ( G ) = p ∗ ω , ω is a K¨ahler metric on M which depends on the Finslermetric G . If the horizontal (1 , -form T satisfies the null eigenvector assumption, then √− ∂ ¯ ∂ log G ( t ) ≥ OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 21 for all t ≥ such that the solution exists. Applications
In this section, we will give two applications of Theorem 2.3.3.1.
The case of curve.
In this subsection, we consider the case of dim M = 1, i.e. M is acurve. In this case, any Hermitian metric ω = √− gdz ∧ d ¯ z on M is K¨ahler automatically. The Gaussian curvature is then given by K = − g ∂ ∂z∂ ¯ z log g = 1 g R g ( ∂∂z , ∂∂ ¯ z , ∂∂z , ∂∂ ¯ z ) =: 1 g R z ¯ zz ¯ z . (3.1)Now we assume that Ω = p ∗ ω + ω F S , (3.2)where ω = ω ( G ) is a metric on M depending smoothly on the Finsler metric G . Then at thepoint ( z , [ v ] , t ), by (2.30), one has( −√− T ( u, u ) = h R g ( u, u ) , − Ψ i Ω − (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) = ( − Ψ) z ¯ z g − R gz ¯ zz ¯ z | u | − f a ¯ b ∂ a Ψ z ¯ z ∂ b Ψ z ¯ z | u | g − = 1 g K ( − Ψ)( u, u ) − f a ¯ b (cid:0) ∂ a (Ψ( U, U )) − i u Ψ ∂ a U (cid:1) ∂ b Ψ z ¯ z g − = 0 , (3.3)since i u Ψ = 0 and ( − Ψ)(
U, U ) attains its local minimal value at the point ( z , [ v ] , t ). There-fore, we prove Proposition 3.1. If M is a curve, then the semipositivity of the curvature of O P ( E ∗ ) (1) ispreserved along the flow (2.48). In particular, if G = h i ¯ j v i ¯ v j comes from a Hermitian metric ( h i ¯ j ) of E ∗ andΩ = p ∗ ω + ω F S (3.4)for a fixed Hermitian metric ω , by Remark 2.1 (1), (2.48) is equivalent to the followingHermitian-Yang-Mills flow: h − · ∂h∂t + Λ R h + ( r − I = 0( h i ¯ j ( t )) > ,h i ¯ j (0) = ( h ) i ¯ j . (3.5)By Proposition 1.8 and Proposition 3.1, we have Corollary 3.2. If M is a curve, then the Griffiths semipositivity is preserved along theHermitian-Yang-Mills flow (3.5). K¨ahler-Ricci flow.
In this section, we assume that E = T M . As the discussion inRemark 2.1 (2), if we take ω ( G ) = √− g α ¯ β dz α ∧ d ¯ z β , where ( g α ¯ β ) denotes the inverse of the matrix (cid:16) ∂ G∂v α ∂ ¯ v β (cid:17) . And G = g α ¯ β v α ¯ v β is a stronglypseudoconvex complex Finsler metric on T ∗ M induced by the following K¨ahler metric ω = √− g ) α ¯ β dz α ∧ d ¯ z β . By (2.5), the flow (2.48) is equivalent to the following K¨ahler-Ricci flow ∂ω∂t + Ric( ω ) + ( n − ω = 0 ,ω > ,ω (0) = ω . (3.6)The solution of (2.48) is induced from the K¨ahler metric ω = √− g α ¯ β dz α ∧ d ¯ z β . In this case, h R g ( u, u ) , − Ψ i Ω = ( − Ψ) α ¯ δ g α ¯ β g γ ¯ δ R gγ ¯ βσ ¯ τ u σ ¯ u τ = 1 G R gµ ¯ να ¯ δ g µ ¯ σ g ¯ ντ v σ v τ g α ¯ β g γ ¯ δ R gγ ¯ βσ ¯ τ u σ ¯ u τ = dim M X α,β =1 R g ( V, V , e α , e β ) R g ( e β , e α , u, u ) , (3.7)where V = √ G g α ¯ σ v σ ∂∂z α and { e α } is a local orthonormal basis of ( T M, ω ). On the other hand,by (2.25), one has at the point ( z , [ v ] , t ), (cid:12)(cid:12) i u ∂ V Ψ (cid:12)(cid:12) = f a ¯ b ∂ a Ψ α ¯ β ∂ b Ψ γ ¯ τ u α u γ g τ ¯ β = 1 G ( R g ) µ ¯ σα ¯ β v σ u α ( R g ) δ ¯ ντ ¯ γ v δ u γ g µ ¯ ν g τ ¯ β = dim M X α,β =1 | R g ( V, e α , u, e β ) | . (3.8)Therefore,( −√− T ( u, u ) = dim M X α,β =1 (cid:16) R g ( V, V , e α , e β ) R g ( e β , e α , u, u ) − | R g ( V, e α , u, e β ) | (cid:17) ≥ Proposition 3.3 ([24, Proposition 1.1]) . If ( M, ω ) is a compact K¨ahler manifold with nonneg-ative holomorphic bisectional curvature, then the nonnegativity is preserved along the K¨ahler-Ricci flow (3.6). OSITIVITY PRESERVING ALONG A FLOW OVER PROJECTIVE BUNDLE 23
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