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 ...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.展开更多
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 ...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.展开更多
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 ...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.展开更多
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 transfer...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.展开更多
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 respectivel...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.展开更多
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.
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.
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...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.展开更多
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 e...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.展开更多
The profile equations of geometric optics are described in a form invariant under the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies f...The profile equations of geometric optics are described in a form invariant under the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies for computing profile equations are equivalent. We prove that if L generates an evolution on L2 the same is true of the profile equations. We prove that the characteristic polynomial of the profile equations is the localization of the characteristic polynomial of the background operator at (y, dφ(y)) where φ is the background phase. We prove that the propagation cones of the profile equations are subsets of the propagation cones of the background operator.展开更多
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is...A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is obtained. Extending the classical notion of exterior diffential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained.展开更多
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 strictl...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.展开更多
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 bundl...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.展开更多
Abstract We study affine Jacobi structures (brackets) on an affine bundle π : A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspon...Abstract We study affine Jacobi structures (brackets) on an affine bundle π : A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A^+ = ∪p∈M Aff(Ap, R) of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A^+. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.展开更多
We give a complete classification of tilting bundles over a weighted projective line of type (2, 3, 3). This yields another realization of the tame concealed algebras of type E6.
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 n...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.展开更多
In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Ein...In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.展开更多
Using the cluster tilting theory,we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of weight type(2,2,2,2).More precisely,a tilting object consisting of rank-two...Using the cluster tilting theory,we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of weight type(2,2,2,2).More precisely,a tilting object consisting of rank-two bundles is constructed via the cluster tilting mutation.Moreover,the cluster tilting approach also provides a new method to classify the endomorphism algebras of the tilting objects in the category of coherent sheaves and the associated bounded derived category.展开更多
A valuable number of works has been published about Hurwitz and Schur polynomials in order to known better their properties. For example it is known that the sets of Hurwitz and Schur polynomials are open and no conve...A valuable number of works has been published about Hurwitz and Schur polynomials in order to known better their properties. For example it is known that the sets of Hurwitz and Schur polynomials are open and no convex sets. Besides, the set of monic Schur polynomials is contractible. Now we study this set using ideas from differential topology, and we prove that the space of Schur complex polynomials with positive leading coefficient, and the space of Hurwitz complex polynomials which leading coefficient having positive real part, have structure of trivial vector bundle, and each space of (Schur complex and real, Hurwitz complex) polynomials has a differential structure diffeomorphic to some known spaces.展开更多
基金supported by Tian Yuan Foundation of China (10526033)China Postdoctoral Science Foundation (2005038639)the Natural Science Foundation of China (10601040,10571144).
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11671330,11688101,11431013).
文摘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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11001146,11101179)the Beijing Higher Education Young Elite Teacher Project
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘The profile equations of geometric optics are described in a form invariant under the natural transformations of first order systems of partial differential equations. This allows us to prove that various strategies for computing profile equations are equivalent. We prove that if L generates an evolution on L2 the same is true of the profile equations. We prove that the characteristic polynomial of the profile equations is the localization of the characteristic polynomial of the background operator at (y, dφ(y)) where φ is the background phase. We prove that the propagation cones of the profile equations are subsets of the propagation cones of the background operator.
文摘A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is obtained. Extending the classical notion of exterior diffential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained.
基金supported by Tsinghua University Initiative Scientific Research Program(Grant No.2019Z07L02016)National Natural Science Foundation of China(Grant No.11901336)。
文摘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.
基金The research was supported by 973 Project Foundation of China and the Outstanding Youth Science Grant of NSFC(grant no.19825105)
文摘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.
基金the Polish Ministry of Scientific Research and Information Technology under the grant No.2 P03A 036 25DGICYT grants BFM2000-0808 and BFM2003-01319D.Iglesias wishes to thank the Spanish Ministry of Education and Cuture for an FPU grant
文摘Abstract We study affine Jacobi structures (brackets) on an affine bundle π : A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A^+ = ∪p∈M Aff(Ap, R) of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A^+. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.
文摘We give a complete classification of tilting bundles over a weighted projective line of type (2, 3, 3). This yields another realization of the tame concealed algebras of type E6.
基金the National Natural Science Foundation of China(Grant No.11671330)。
文摘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.
基金supported in part by National Natural Science Foundation of China (Grant No. 10901147)supported in part by National Natural Science Foundation of China (Grant Nos. 10831008 and 11071212)the Ministry of Education Doctoral Fund 20060335133
文摘In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.
基金supported by National Natural Science Foundation of China(Grant Nos.11571286,11871404 and 11801473)the Fundamental Research Funds for the Central Universities of China(Grant Nos.20720180002 and 20720180006)。
文摘Using the cluster tilting theory,we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of weight type(2,2,2,2).More precisely,a tilting object consisting of rank-two bundles is constructed via the cluster tilting mutation.Moreover,the cluster tilting approach also provides a new method to classify the endomorphism algebras of the tilting objects in the category of coherent sheaves and the associated bounded derived category.
文摘A valuable number of works has been published about Hurwitz and Schur polynomials in order to known better their properties. For example it is known that the sets of Hurwitz and Schur polynomials are open and no convex sets. Besides, the set of monic Schur polynomials is contractible. Now we study this set using ideas from differential topology, and we prove that the space of Schur complex polynomials with positive leading coefficient, and the space of Hurwitz complex polynomials which leading coefficient having positive real part, have structure of trivial vector bundle, and each space of (Schur complex and real, Hurwitz complex) polynomials has a differential structure diffeomorphic to some known spaces.