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.     

On the restriction of vector bundles to good families of curves

Summary Here we study the restriction to suitable curves of certain vector bundles on projective varieties. We handle in particular: (a) the descent of a vector bundle for the contraction of an extremal ray on a smooth 3-fold (using Mori theory); (b) the stability or semistability (in any characteristic) of certain homogeneus vector bundles on Pⁿ and their restriction to suitable subvarieties.     

Regulous vector bundles

Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified‐algebraic vector bundles. Furthermore, using new results on curve‐rational functions, we characterize regulous vector bundles among families of vector spaces parametrized by an affine regulous variety. We also study relationships between regulous and topological vector bundles.     

Holomorphic vector bundles on non-algebraic surfacesFibrés vectoriels holomorphes sur les surfaces non algébriques

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is, in general, still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces. To cite this article: A. Teleman, M. Toma, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 383–388.     

Quaternionic analytic torsion

We define an (equivariant) quaternionic analytic torsion for anti-self-dual vector bundles on quaternionic Kähler manifolds, using ideas by Leung and Yi. We do so by constructing a Laplace operator associated to a complex defined by Salamon. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl groups.     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

Brill-Noether theory on Hirzebruch surfaces

In this paper, we investigate higher rank Brill-Noether problems for stable vector bundles on Hirzebruch surfaces. Using suitable non-splitting extensions, we deal with the non-emptiness. Results concerning the emptiness follow as a consequence of a generalization of Clifford’s theorem for line bundles on curves to vector bundles on surfaces.     

本刊英文版Vol.30(2014),No.10论文摘要(英文)

<正>on Double Vector Bundles Zhuo CHEN Zhang Ju LIU Yun He SHENG Abstract 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     

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.     

Chern classes of tropical vector bundles

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in Torchiani, C., Line Bundles on Tropical Varieties, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern, 2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Verlinde bundles and generalized theta linear series

In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties.

We study a naturally defined class of vector bundles on a Jacobian, called Verlinde bundles, in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.

     

Quasi-parabolic Siegel formula

The result of Siegel that the Tamagawa number ofSL _r over a function field is 1 has an expression purely in terms of vector bundles on a curve, which is known as the Siegel formula. We prove an analogous formula for vector bundles with quasi-parabolic structures. This formula can be used to calculate the Betti numbers of the moduli of parabolic vector bundles using the Weil conjuctures An erratum to this article is available at .     

Interpolation Problems for Vector Bundles on Algebraic Curves

In this paper we study meromorphic maps between vector bundles on a Riemann surface. We are mainly interested in stable vector bundles. For a huge number of numerical data we prove the existence of a meromorphic map between two vector bundles with a prescribed number of zeroes and a prescribed number of poles.     

p-adic representations and vector bundles on Mumford curves

Christopher Deninger andAnnette Werner constructed a functor that associates representations of the algebraic fundamental group of an algebraic curve to a class of vector bundles on that curve. We compare this to a construction byFaltings for Mumford curves that associates representations of the Schottky group to semistable vector bundles of degree 0. We prove that for a certain class of vector bundles on Mumford curves the constructions induce isomorphic representations.     

A corollary to Kodaira-Spencer’s theorem on continuity of eigenvalues

We give an elementary proof of continuity of the determinant in the parameter for a smooth family of laplacians (of the same nullity) on a smooth family of holomorphic vector bundles over a compact complex manifold. Families of unitary flat bundles over a compact Riemann surface are discussed, as an example.     

Nuclear dimension, $$\mathcal{ }$$-stability,and algebraic simplicity for stably projectionless $$C^*$$C∗-algebras

Procesi bundles are certain vector bundles on symplectic resolutions of quotient singularities for wreath-products of the symmetric groups with the Kleinian groups. Roughly speaking, we can define Procesi bundles as bundles on resolutions that provide derived McKay equivalence. In this paper we classify Procesi bundles on resolutions obtained by Hamiltonian reduction and relate the Procesi bundles to the tautological bundles on the resolutions. Our proofs are based on deformation arguments and a connection of Procesi bundles with symplectic reflection algebras.     

Vector bundles on p-adic curves and parallel transport

We define functorial isomorphisms of parallel transport along étale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.     

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld–Mukai bundles on smooth projective surfaces X using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld–Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of X corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld–Mukai bundles. This gives semistable Lazarsfeld–Mukai bundles on Kawamata covers of the projective plane and of certain K3 surfaces.     

Parameters of AG codes from vector bundles

We investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples.