Bochner-Kodaira Techniques on Kähler Finsler Manifolds∗

Jinxiu XIAOChunhui QIUTongde ZHONG

1 Introduction

Bochner[1–3]initiated a method,i.e.,the well-known “Bochner technique”,which used the Laplace operator and the general maximum principle of Hopf to deal with the relation between vector or tensor fields and the curvature of manifolds,and got the global properties of manifolds.From then on,the Bochner technique became a very useful method in geometrical study.Both in Riemannian and Kählerian manifolds,the Bochner technique was discussed in details in[4–8].The Bochner technique is used to integrate the Laplacian of the pointwise square norm of a harmonic form over a compact Riemannian manifolds,yielding thereby two terms.One is the global square norm of the covariant derivatives of the harmonic form.The other involves the curvature tensor.Under the suitable condition of the curvature tensor,it can be obtained that the harmonic form must be zero or parallel.It was applied to(0,q)-forms on a Kähler manifold with values in Hermitian holomorphic line bundles,due to Kodaira[9].Later,the technique was called as the Bochner-Kodaira technique.The Bochner-Kodaira technique is the important method in differential geometry and is variated as the∂∂Bochner-Kodaira technique due to Siu[10–11].

Recently,under the initiation of S.S.Chern,the global differential geometry of real and complex Finsler manifolds gained a great development(see[12–17]),Abate and Pateizio[16]set up a Cartan-Finsler connection in real Finsler manifolds and a Chern-Finsler connection in complex Finsler manifolds.The main purpose of this paper is to generalize the Bochner-Kodaira techniques from Kähler manifolds to Kähler Finsler manifolds,and the vanishing theorem is obtained by using theBochner-Kodaira technique.

2 Complex Finsler Manifolds

LetMbe a compact complex manifold of dimensionn,and,whereT1,0Mis the holomorphic tangent bundle ofM.We denote bythe zero section ofT1,0M,and setwhich means the holomorphic tangent bundle minus its zero section.Letandbe the local coordinates onMand the induced complex coordinates onT1,0M,respectively.For simplicity,we denote

which give a local holomorphic frame field of

Since there is a naturalacting onTMby scalar multiplication,the projective tangent bundle PTM can be defined byC∗}.The local coordinate system(z,v)for TM may also be considered as a local coordinate system for PTM as long asvis considered as a homogeneous coordinate system.The reason for working on PTM rather than TM is that PTM is compact,ifMis compact.

LetFbe a strongly pseudoconvex complex Finsler metric defined onT1,0M,that isF:is a continuous function satisfying the following conditions(see[16]):

(i)G=F2is smooth on

(ii)F(v)>0 for all

(iii)for allandλ∈C;

(iv)The Hermitian matrix(Gαβ)is positive de finite on,where

and the derivatives with respect to thez-coordinates will be denoted by indexes after a semicolon,for instance,

A manifoldMendowed with a strongly pseudoconvex complex Finsler metric will be called a strongly pseudoconvex complex Finsler manifold.

Letbe the complexity of the real tangent bundleandTCMbe the complexity of the real tangent bundleTRMofM.Then the differential dπ:defines the vertical bundleVover?Mby

which is a holomorphic vector bundle of ranknoverA local frame field ofVis given byand there is a well-defined Hermitian metric onVinduced byFgiven by

whereThen there is a unique Chern-Finsler connectionDassociated to the Hermitian structure induced byF.Then the Chern

Finsler connection is the Hermitian connection of the holomorphic vector bundle(V?,?).Letbe the complex horizontal bundle associated to the Chern Finsler connection,and the natural local frame{δ1,···,δn}forHis given by

whereare called the nonlinear connection coefficients associated to(M,F).In this paper,we shall only use the adapted frameand its dual framewherebecause they have a simple rule of transformation under the change of coordinates.

Using the complex horizontal map Θ :V→H,the Hermitian metric?,?onVcan be transferred onHby settingforandH1,H2∈Hv.Note that we shall userather thanfor simplicity,when there is no chance of confusion.The Hermitian metric?,?onis defined by requiringHto be orthogonal toVand the Chern Finsler connection extends to the complex linear connection still called the Chern Finsler connection onwhich is compatible with the Hermitian metriconbut is not torsion free in general.It hasH-valued(2,0)-torsionθandV-valued(1,1)-torsionτ,andθrelates to the Kählerianity of the Chern Finsler connectionD.More precisely,a strongly pseudoconvex complex Finsler metricFis called strongly Kähler if and only if

called Kähler if and only if

and called weakly Kähler if and only if

Recently,Chen and Shen[18]showed that a Kähler-Finsler metric must be a strongly Kähler-Finsler metric.Then,it is necessary to consider the Kählerian case in this paper.

By definingand the complex linearity,the Chern Finsler connectionDcan be extended to the whole complex vector bundleand its dual complex vector bundleby requiringfor everyandThus the Chern Finsler connection can also be extended to the complex linear connectionin the usual way.All the extended connections are still called the Chern Finsler connection with the conjugation and preserving the type.Let∇be the covariant differentiation defined byD.Since the complex Finsler fundamental tensorGαβis bothH-metrical andV-metrical,i.e.,

Gαβare also bothH-metrical andV-metrical.

3 Bochner-Kodaira Techniques for the Pull-Back Bundles of Holomorphic Vector Bundles

The principal step in the Bochner-Kodaira technique is the computation of the Laplacian.There are some results about the Laplacian and their applications for horizontal(p,q)-forms on the base manifold or the tangent bundle(see[19–23]).In preparation,we will give simple statements for the Laplacian for the horizontal(p,q)-forms on PTM,and omit the complicated computation.In this section,we focus on the horizontal Laplacian?Hof the horizontal(p,q)-form with the value in the Hermitian holomorphic vector bundles on PTM.Then,we will give the expression of?Hexplicitly in terms of the horizontal covariant derivatives of the Chern Finsler connection,which was called the Bochner-Kodaira technique.

Let(M,F)be a compact Kähler Finsler manifold.It is known thatFinduces naturally a non-degenerated Hermitian metric on the total space PTM,

Denote

Then the invariant volume form of PTM is given by

If we denote by dσthe pure vertical form of the volume form of PTM,thus

and then

LetAp,qbe the space of horizontal(p,q)-forms on PTM,that is,those coefficients of everyϕ∈Ap,qare zero homogeneous with respect to fibre coordinates,and the elements ofAp,qin local coordinates are

Then the pointwise inner product is given by

If we denote

whereis a permutation of(1,2,···,n).Similarly,forBq=(β1,···,,and

then

Notice that there is a global inner product inAp,qgiven by

Then we can define the operatorby the relation

It is easy to obtain that the operator∗has the following properties:

where

If(M,F)is a Kähler Finsler manifold,then by the symmetry of the horizontal connection coefficients:the horizontal derivatives can be replaced by the horizontal covariant derivatives,that is,

Letandbe the adjoint operators ofandwith respect to the global inner product inAp,q,respectively,that is,

which satisfyandThen by solving,one can get

By de fi ning the horizontal Laplacian operator

we can get its expression explicitly in terms of the horizontal covariant derivatives of the Chern Finsler connection.

Theorem 3.1(see[21])For any ϕ∈Ap,q,

In the following,we emphasize on deriving the horizontal Laplacian operator for the horizontal(p,q)-form with value in the Hermitian holomorphic vector bundle on PTM by extending the operators,without confusion,and we continue to use all the symbols,respectively.

where

The coefficients of the horizontal part ofare

so we can de fi ne a horizontal curvature form ofby

If the strongly pseudoconvex complex Finsler manifold is a Kähler manifold,then the nonlinear connectionand the horizontal curvature form(3.12)reduces to an ordinary curvature form of the Kähler manifold(see[10]).Furthermore,for a Kähler Finsler manifold,there exists a normal coordinate system(see[24–25]),for which one also has the nonlinear connectionand in this case,the horizontal curvature form(3.12)takes the form of a Kähler manifold,that is,in the normal coordinate system,the Kähler Finsler manifold is very similar to a Kähler manifold,so one often uses the normal coordinate system to simplify calculations(see[5]).

Letbe the space of complex horizontal(p,q)-forms on PTM with value inIfis a local holomorphic frame ofthenare

respectively,where,and we have

Then we define

whereare the horizontal covariant derivatives of the Chern Finsler connection.

We can define the pointwise inner product onby

whereis defined by(3.1).

The global inner product can be defined by

and we defineas usual.

The adjointofoncan be defined by solving

forandand then

but

Thus

Then

In order to obtain the expression ofexplicitly,we need to computeForwe can write

and

If we set

then

and

whereEq−1is the increasing set of numbers complementary to the set.Letbe the increasing set of numbers complementary to the setand set

Then

and

by which,and(3.18),we get

Hence,we have the following result.

Proposition 3.1For any

By defining the horizontal Laplacian operator?Hfor the holomorphic vector bundle on PTM,we have

and?Hϕ=0forif and only if ϕ is harmonic horizontal(p,q)-form with value in

Theorem 3.2If(M,F)is a Kähler Finsler manifold,for anywe have

ProofFrom(3.3)and(3.13),

Then

and

Thus

and

Lemma 3.1(see[16])Let be the complex linear connection oninduced by the Chern-Finsler connection.Then for anywe have

whereΩis the curvature operator of the Chern Finsler connection D.In local coordinates,the curvature operator is given by

and

Let us calculate the second term on the right-hand side of(3.21).For a formϕαof type(1,0),

For a formϕβof type(0,1),

Similarly,we see

Then,(3.21)can be rewritten as

Let,whereis understood in the sense of(3.15).WhenMis a compact Kähler Finsler manifold,by Stokes’theorem,

wheredenotes the globalL2norm over PTM,anddenotes the-valued tensor with components.(3.27)is theBochner-Kodaira technique.

Since

that is,

by applying the commutation formula forto(3.21),with(3.25),we obtain

WhenMis a compact Kähler Finsler manifold,and by contracting(3.29)withand integrating over PTM,we obtain

wheredenotes the globalL2norm over PTM,anddenotes the-valued tensor with components(3.30)is theBochner-Kodaira technique.

Remark 3.1If the Kähler Finsler manifold is a Kähler manifold,then(3.27)and(3.30)coincide with(1.3.3)and(1.3.5)in[11].

4 Bochner-Kodaira Technique

Bochner-Kodaira technique was initiated by Siu[10]and named in[11],and the method is a modi fi cation of the classical Bochner-Kodaira technique by replacing the operator?withand exploits the bigraded structure of the differential forms on a Kähler manifold in a more serious way than usual in the computations with?.In this section,we study theBochner-Kodaira technique on Kähler Finsler manifolds and get the vanishing theorem for the Hermitian holomorphic bundle on PTM on Kähler Finsler manifolds.

Letbe the Hermitian holomorphic vector bundle of rankron PTM is the same bundle as stated in Section 3.We know that there exist normal coordinates ofMand normal fiber coordinates ofwhich can be found with detailed information in[24–25].This greatly simplifies our calculations in local coordinates,and then,in all the following computations,we will use the normal coordinates ofMand the normal fiber coordinates of

WhenMis a compact Kähler Finsler manifold,and by integrating over PTM,we obtain

By applying integration by parts to the second term and the last term,from

it follows that

Likewise,

Hence

We are going to transform,by using the exterior algebra of Hermitian vector spaces,each term in(4.4)to a corresponding term obtained from theBochner-Kodaira technique.

For any primitivek-formψands≤r,

for 0≤l≤n−k.One has

Everyk-formvcan be uniquely written as

Lemma 4.1(see[11])For any(1,q)-form η,

For detailed information of these formulae,see[6,p.69].

Lemma 4.2

ProofSinceϕλis(0,q)-form,it is primitive.By(4.6),

Using local coordinates,we have

Contracting both sides withGαβ,we obtain

Taking the inner product withϕλ,we obtain the desired equation.

By applying(4.6)to the caseandl=0,we get

By using Lemma 4.1 in the casewe have

which is the same as(3.30)obtained byBochner-Kodaira technique under the normal coordinates ofMand the normal fiber coordinates of

Then combining(4.4)with(4.9)–(4.10),we have

Then,we have the following result.

then there is no nonzero horizontal harmonic(0,q)-form over PTM with valued infor all0

ProofSinceMis compact,ϕis horizontal harmonic if and only ifFrom(4.12),we have

which contradicts(4.13)whenϕis not identically zero.Henceϕ≡0.

[1]Bochner,S.,Vector fields and Ricci curvature,Bull.Amer.Math.Soc.,52,1946,776–797.

[2]Bochner,S.,Curvature in Hermitian metric,Bull.Amer.Math.Soc.,53,1947,179–195.

[3]Bochner,S.,Curvature and Betti numbers,I,II,Ann.of Math.,49,1948,379–390;50,1949,77–93.

[4]Yano,K.and Bochner,S.,Curvature and Betti Numbers,Princeton Univ.Press,Princeton,1953.

[5]Wu,H.H.,The Bochner Technique in differential Geometry,Harwood Academic Publishers,London,Paris,1988.

[6]Morrow,J.and Kodaira,K.,Complex Manifold,Holt,Rinehart and Winston,New York,1971.

[7]Griffiths,P.and Harris,J.,Principles of Algebraic Geometry,John Wiley and Sons Inc.,New York,1978.

[8]Kobayashi,S.,differential Geometry of Complex Vector Bundles,Princeton Univ.Press,New Jersey,1987.

[9]Kodaira K.,On a differential-geometric method in the theory of analytic stacks,Proc.Nat.Acad.Sci.,39,1953,1268–1273.

[10]Siu,Y.T.,The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds,Ann.of Math.,112,1980,73–111.

[11]Siu,Y.T.,Complex analyticity of harmonic maps,vanishing theorems and Lefchetz theorems,J.Di ff.Geom.,17,1982,55–138.

[12]Bao,D.,Chern,S.S.and Shen,Z.,An Introducetion to Riemann-Finsler Geometry,Springer-Verlag,New York,2000.

[13]Chern,S.S.,Finsler geometry is just Riemannian geometry without the quadratic restriction,AMS Notices,43(9),1996,959–963.

[14]Bao,D.,Chern,S.S.and Shen,Z.,Finsler Geometry(Proceedings of the Joint Summer Research Conference on Finsler Geometry,July 16–20,1995,Seattle,Washington),Cont.Math.,Vol.196,A.M.S.,Providence,RI,1996.

[15]Shen,Z.,Lectures on Finsler Geometry,World Scientific Publishers,Singapore,2001.

[16]Abate,M.and Patrizio,G.,Finsler metric—A global approach,Lecture Notes in Math,Vol.1591,Springer-Verlag,Berlin,1994.

[17]Wong,P.M.,A survey of complex Finsler geometry,Adv.Stud.Pure Math.,48,2007,375–433.

[18]Chen,B.,Shen,Y.,Kähler Finsler metric are actually strongly Kähler,Chin.Ann.Math.,30B(2),2009,173–178.

[19]Bao,D.and Lackey,B.,A Hodge decomposition theorem for Finsler spaces,C.R.Acad.Sci.Paris,323(1),1996,51–56.

[20]Antonelli,P.and Bao,D.,Finslerian Laplacians and Applications,MAIA 459,Kluwer Academic Publishers,Dordrecht,1998.

[21]Zhong,C.and Zhong,T.,Horizontalα-Laplacian on complex Finsler manifolds,Science in China,Ser.A,48(supp.),2005,377–391.

[22]Zhong,C.and Zhong,T.,Hodge decomposition theorem on strongly Kähler Finsler manifolds,Science in China,Ser.A,49(11),2006,1696–1714.

[23]Zhong,C.,Laplacians on the holomorphic tangent bundles of a Kähler manifolds,Science in China,Ser.A,52(12),2009,2841–2854.

[24]Aikou,T.,On complex Finsler manifolds,Rep.Fac.Kagoshima Univ.,35,1991,9–25.

[25]Xiao,J.,Zhong,T.and Qiu,C.,Bochner technique in Strongly Kähler Finsler manifolds,Acta Math.Sci.,30B(1),2010,89–106.