We consider a vector bundle E → M and the principal bundle PE of frames of E. Let K be a principal connection on PE and let Λ be a linear connection on M . We classify all principal connections on W 2PE = P 2M ×M J2PE naturally given by K and Λ.

Let T : M → M be an invertible measurable transformation, and μ be a T -invariant probability on M . Let π : E → M be a finite-dimensional real vector bundle over M , endowed with a measurable Riemannian metric ‖ · ‖. Let F : E → E be a measurable vector bundle automorphism over T : this means that π ◦ F = T ◦ π and the action Fx : Ex → ET (x) of F on each fiber Ex = π−1(x) is a linear isomorph...

Let L be a (semi)-positive line bundle over a Kähler manifold, X , fibered over a complex manifold Y . Assuming the fibers are compact and non-singular we prove that the hermitian vector bundle E whose fibers are the space of global sections to L⊗KX/Y endowed with the L-metric is (semi)-positive in the sense of Nakano. As an application we prove a partial result on a conjecture of Griffiths on ...

In this paper, we prove a generalization of Reznikov's theorem on the torsion-property of the Chern-Simons classes and in particular the torsion–property of the Deligne Chern classes (in degrees > 1) of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi–projective variety with an ir-reducible smooth divisor at infinity. We define the Chern-Simons...

We generalize Horrocks’ criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on arbitrary smooth complex projective varieties of dimension ≥ 4, which asserts that a vector bundle E on X splits iff its restriction E|Y to an ample smooth codimension 1 subvariety Y ⊂ X splits.

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classificati...

This paper deals with Arakelov vector bundles over an arithmetic curve, i.e. over the set of places of a number field. The main result is that for each semistable bundle E, there is a bundle F such that E⊗F has at least a certain slope, but no global sections. It is motivated by an analogous theorem of Faltings for vector bundles over algebraic curves and contains the Minkowski-Hlawka theorem o...

where fc is the complexification of the bundle f (see [4, XII] for details). Thus, pi(Z)EH"(B; Z), where B is the base of j\ For the results which follow we will need the following relationship between the Pontrjagin and Chern classes. Suppose thatco" is a complex re-plane bundle. Each fibre of wM is then an re-dimensional complex vector space. By restricting attention to the real numbers as sc...

Let G ~ Sp (n, R) be the symplectic group (n~i) and G 6 be the same group with the discrete topology. The present paper is devoted to the study of the element u of the group H ~ (G~; Z), which is determined up to sign in any of the following (equivalent) ways: a) u is the image under the canonical homomorphism H 2 (BG; Z)-~ H 2 (BG~: Z) of the generator of the group H 2 (BG; Z)~ Z; b# u is the ...

Unexpected complex of jumping lines We prove here the following results : Theorem 1 Let E a rank 2 vector bundle over P3, if C is a reduced irreducible curve of P3 such that EH is unstable for all H ∈ C then C is a line. We define now the set W (E) as the set of planes H such that the restricted bundle EH is unstable (that means non semi-stable). Theorem 2 Let E a rank 2 vector bundle over P3, ...