HORIZONTAL LAPLACE OPERATOR IN REAL FINSLER VECTOR BUNDLES

A vector bundle F over the tangent bundle TM of a manifold M is said to be a Finsler vector bundle if it is isomorphic to the pull-back π^＊E of a vector bundle E over M（[1]）. In this article the authors study the h-Laplace operator in Finsler vector bundles. An h-Laplace operator is defined, first for functions and then for horizontal Finsler forms on E. Using the h-Laplace operator, the authors define the h-harmonic function and ho harmonic horizontal Finsler vector fields, and furthermore prove some integral formulas for the h-Laplace operator, horizontal Finsler vector fields, and scalar fields on E.

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS(Ⅰ)

This paper is a continuation of a previous one. We still emphasize the discussionon the relation between the dynamics on the base space of a rector bundle and that oneach associated bundle of frames.

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS(Ⅱ)──PART 1

The study of linear and global. properties of linear dynamical systems on vector bundles appeared rather extensive already in the past.Presently we propose to study perturbations of this linear dynamics The perturbed dynamical system which we shallconsider is no longer linear.while the properties to be studied will be still global in general.Moreover.we are interested in the nonuniformly hyperbolic properties.In this paper,we set an appropriate definition for such perturbations.Though it appearssome what not quite usual yet has deeper root in standard systens of differential equations in the theory of differentiable dynamical systens The general problen is to see which property of the original given by the dynamical system is persistent when a perturbation takes place.The whole contenl of the paper is deyoted to establishinga theorem of this sort.

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS (Ⅱ)──PART 2

In the part 2, theorem 3.1 stut ed in part 1[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differentialequations to a form concerning Standard systems of equations in the theory ofdifferentiable dynamical systems. Then by using theorem 3.1 together with thepreliminary theorem 2.l, foe main theorem of this paper announced in part 1 is proved.The definition of admissible perturbation is contained in the appendix of part 2. Themeanings of the main theorem is described in the introduction of part 1.

On Orthogonal Decompositions of "Pull Back" Hermitian Holomorphic Vector Bundles

Let f ： Ω→Gr（n,H） be a holomorphic curve, where Ω is a bounded open simple connected domain on the complex plane C and Gr（n,H） the Grassmannian manifold. Denote by Ef the ＂pull back＂ bundle induced by f. We show the uniqueness of the orthogonal decomposition for those complex bundles. As a direct application, we give a complete description of the HIR decomposition of a Cowen- Douglas operator T ∈ Bn（Ω）. Moreover, we compute the maximal self-adjoint subalgebra of A＇（Ef） and A＇（T） respectively. Finally, we fix the masa of A＇（Ef） and .A＇ （T） which depends on the HIR decomposition of Ef or T respectively.

Weakly complex Einstein-Finsler vector bundle Dedicated to Professor Tongde Zhong for His 90th Birthday

Let （E, F） be a complex Finsler vector bundle over a compact Kahler manifold （M, g） with Kahler form φ We prove that if （E, F） is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou （1997）, then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle （E, F） over a compact Kahler manifold （M, g） is necessarily φ-semistable and （E,F）=（E1,F1）……（Ek,Fk）,where Fj ：= F｜Ej, and each （Ej, Fj） is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a φ-stable Einstein-Hermitian vector bundle with the same factor c as （E, F）.

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vector bundle of rank r with trivial pull-back to W=■~n-{0}.We prove the existence of a non-vanishing section of L■E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D.Mall about structure theorem of such a vector bundle E.

Some results about holomorphic vector bundles over general Hopf manifolds

In the present note that grew out of my talk given at the conference in honor of Prof. Zhong Tongde,I give a survey of some recent results about holomorphic vector bundles over general Hopf manifolds.

Brill-Noether Theory for Rank Two Vector Bundles Generated by Their Sections

We here study the Brill-Noether theory for rank two vector bundles generated by their sections. We generalize the vanishing theorem, the Clifford theorem and the existence theorem to such bundles.

SUB-SIGNATURE OPERATORS,η-INVARIANTS AND A RIEMANN-ROCH THEOREM FOR FLAT VECTOR BUNDLES

The author presents an extension of the Atiyah-Patodi-Singer invariant for unitary representations [2,3] to the non-unitary case, as well as to the case where the base manifold admits certain finer structures. In particular, when the base manifold has a fibration structure, a Riemann-Roch theorem for these invariants is established by computing the adiabatic limits of the associated η-invariants.

The Total Stiefel-Whitney Classes of Vector Bundles on CP（n） × CP（m）

The possible form of the total Stiefel-Whitney classes of vector bundles on CP(n)× CP(m) is determined in this paper.

The Structure of Vector Bundles on Non-primary Hopf Manifolds

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X,with trivial pull-back to C^n-{0}.The authors show that there exists a line bundle L over X such that E■L has a nowhere vanishing section.It is proved that in case dim(X)≥3,π*(E)is trivial if and only if E is filtrable by vector bundles.With the structure theorem,the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E.

On the Cayley-Bacharach property and the construction of vector bundles

We study the Cayley-Bacharach property on complex projective smooth varieties of dimension n≥2 for zero dimensional subscheme defined by the zero set of the wedge of r-n + 1 global sections of a rank r≥n vector bundle,and give a construction of high rank reflexive sheaves and vector bundles from codimension 2 subschemes.

On Double Vector Bundles

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theory of double vector bundles in term of the corresponding short exact sequences. Examples including the jet bundle and the Atiyah algebroid are discussed.

On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n,and let E be an ample vector bundle over X.We show that any Schur class of E,lying in the cohomology group of bidegree(n-1,n-1),has a representative which is strictly positive in the sense of smooth forms.This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level,and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld(1983)for certain degrees.

Some Results on Special Stable Vector Bundles of Rank 3 on Algebraic Curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 2 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3^2,d and w3 respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3^2,4 and a complete discussion on the structure of ω3^3,2/3g＋2.

Kahler Metrics on the Projective Bundle of a Holomorphic Finsler Vector Bundle

Let(M,g)be a compact Kihler manifold and(E,F)be a holomorphic Finsler vector bundle of rank r≥2 over M.In this paper,we prove that there exists a Kahler metricФdefined on the projective bundle P(E)of E,which comes naturally from g and F.Moreover,a necessary and sufficient condition forФhaving positive scalar curvature is obtained,and a sufficient condition forФhaving positive Ricci curvature is established.

BRILL-NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Let X be an arbitrary smooth irreducible complex projective curve, E (?)X a rank two vector bundle generated by its sections. The author first represents ?as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H?X, [D1] |D2) is a. collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using / and the Brill-Noether matrix of D1+D2, the author constructs a 2g × d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?) Im{H?X, E) (?) H0(X,[D2]) (?) H0(X, [D2]|D2)} From this and H0(X,E) = H0(X,[D1]) (?)Im{H0(X,E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.

Globally generated vector bundles on a smooth quadric surface

We give the classification of globally generated vector bundles of rank 2 on a smooth quadric surface with c1(2,2)in terms of the indices of the bundles,and extend the result to arbitrary higher rank case.We also investigate their indecomposability and give the sufficient and necessary condition on numeric data of vector bundles for indecomposability.

THE FOURIER TRANSFORM FOR HOMOGENEOUS VECTOR BUNDLES OVER QUATERNION UNIT DISK

The Fourier transform for homogeneous vector bundles over quaternion unit disk is studied, and the corresponding inversion formula and Plancherel formula are established.