On stable and uniform rank-2 vector bundles on P2 in characteristic p

The uniform rank-2 vector bundles on Pⁿ are determined and the behaviour of the stable rank-2 vector bundles on P² under restriction to a general line is studied, where Pⁿ denotes the n-dimensional projective space over an algebraically closed field of positive characteristic.     

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.     

On Framed Instanton Bundles and Their Deformations

We consider a compact twistor space P and assume that there is a surface S ⊂ P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space ([20], [18]). Similar to [6] and [5] we examine the restriction of an instanton bundle V equipped with a fixed trivialization along F to a framed vector bundle over (S, F). First we develope inspired by [13] a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)‐instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)‐instanton bundles and isomorphism classes of framed vector bundles over (S, F) due to [5] is actually an isomorphism of moduli spaces.     

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.     

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.     

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.     

Rank 4 vector bundles on the quintic threefold

By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P ⁴ the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext ¹ (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.     

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.     

Bordisme de S1-fibrés vectoriels munis¶de structures supplémentaires

Bordism of S ¹-vector bundles with additional structures We give isomorphisms between equivariant bordism groups of certain S ¹-vector bundles and bordism groups of suitable “classifying” spaces determined by certain caracterestic classes. In the spinorial case, we detect the even or odd type of the S ¹-action and give a relationship with elleptic homology. Furthermore, we define a new type of $S^1$-actions, depending on the actions and the given slice type. This new type differs, in certain cases, from the classical odd or even type of S ¹-actions on spinorial manifolds. Received: 7 July 2000 / Revised version: 10 February 2001     

Grauert- and Lax-Halmos-type theorems and extension of matrices with entries in H

In the paper we prove an extension theorem for matrices with entries in H^∞(U) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras.     

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

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.     

ACM vector bundles on scrolls

Here we show that certain low rank ACM vector bundles on scrolls over smooth curves are iterated extensions of line bundles. Partially supported by MIUR and GNSAGA of INDAM (Italy)     

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C^∗-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.     

Mumford's example and a general construction

We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.     

Torus Bundles over Locally Symmetric Varieties Associated to Cocycles of Discrete Groups

 We construct torus bundles over locally symmetric varieties associated to cocycles in the cohomology group , where Γ is a discrete subgroup of a semisimple Lie group and L is a lattice in a real vector space. We prove that such a torus bundle has a canonical complex structure and that the space of holomorphic forms of the highest degree on a fiber product of such bundles is isomorphic to the space of mixed automorphic forms of a certain type. (Received 4 September 1998)     

Quantification de la dualité de Grassmann

Let V be a complex vector space of dimension n, %plane1D;53E; (resp. %plane1D;53E;^*) the Grassmann manifold of p-dimensional (resp. (n — p)-dimensional) subspaces of V, and of Ω the relation of transversality in %plane1D;53E;*%plane1D;53E;^*. We announced in [6] equivalences between derived categories of sheaves and of D-modules on %plane1D;53E; and %plane1D;53E; defined by the integral transforms associated to Ω. We show here that these transforms exchange the D-modules associated to the holomorphic line bundles on %plane1D;53E; and %plane1D;53E;^*. This is equivalent to “quantizing” the underlying contact transformation between certain open dense subsets of the cotangent bundles. In the case p = 1, we recover already known results for the projective duality (see [1] and [5]).     

Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on \mathbb{C}{{\mathbb{P}}^1}

Consider the moduli space of parabolic Higgs bundles (E, Φ) of rank two on ??¹ such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by $ \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) $ . We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.     

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy-Riemann equations on affine spaces.

     

Strange bundles onP 2 and elementary transformations

Summary Here we begin the study of moduli of vector bundles on a surfaceS with a fixed restriction to a divisorD. Here we stress the caseD≊P ¹. In this way we construct many families of stable rank-2 bundles onP ² with unbalanced general splitting type (in characteristicp>0).

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Riassunto Si comincia qui lo studio dei moduli di fibrati vettoriali su una superficieS con una assegnata restrizione ad un divisoreD (quasi sempre qui conD≊P ¹). In caratteristicap si ottengono così molte famiglie di fibrati stabili suP ² con ?strana? restrizione ad una retta generica.