This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-WeilSimons theory for smooth bundle maps α : E → F which, for smooth connections on E and F , establishes formulas of the type

We give a class of examples of a vector bundle on a relative smooth projective curve over SpecZ such that for infinitely many prime reductions the bundle has a Frobenius descent, but the generic restriction in characteristic zero is not semistable. Mathematical Subject Classification (2000): primary: 14H60, secondary: 13A35.

Given an action of a compact group on complex vector bundle, there is induced the associated Cuntz--Pimsner algebra. We determine conditions under which this has finite Rokhlin dimension.

Let G := Z/nZ, n ≥ 2, act diagonally on C and set X := C 2 \ {0}/G (a punctured neighborhood for the normal surface singularity An−1). Here we prove the existence of a rank two holomorphic vector bundle on X without rank one subsheaves. 1. Unfiltrable Vector Bundles Let M be a reduced and irreducible complex space and E a rank two holomorphic vector bundle on M . We will say that E is unfiltrab...

∇ : V → V ⊗ ΩX . Then the flat sections of V determine a local system L on X. For every point x ∈ X, the fiber of the local system Lx can be identified with the fiber Vx. Given a path p : [0, 1] from x = p(0) to y = p(1), there is a map p! : Lx → Ly is given by parallel transport along p, using the connection ∇; moreover the map p! depends only on the homotopy class of the path p. This construc...

We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. characterize space on a bundle as an affine modeled over vector $$1$$ -forms values cocycles in algebra aforementioned bundle. show satisfy Ambrose–Singer theorem bundles can be seen particular case associated realizing connections. give construction Moduli fixed base fiber, r...

— We consider linear codes defined by the projective bundle associated to a vector bundle over a curve. We prove the existence of codes with explicit parameters. Résumé (Codes de Reed-Muller relatifs sur des fibrés projectifs). — Nous étudions les codes linéaires définis par le fibré projectif associé à un fibré vectoriel sur une courbe. Nous prouvons l’existence de codes avec des paramètres ex...

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called jkj{grading, i.e. a grading of the form g = g ?k g k , such that no simple factor of G is of type A 1. Let P be the subgroup corresponding to the subalgebra p = g 0 g k. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G; P) and to study ...

Let X be a smooth projective curve of genus g ≥ 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend’s conjecture on the canonical reduction of principal G-bundles for G = SO(n) with n ≥ 7. Let X be a smooth projective curve of genus g ≥ 2 and let G be a connected reduc...