In this article, we generalize the concept of the Lie algebra of vector fields to the set of smooth sections of a T-bundle which is by definition a canonical generalization of the concept of a tangent bundle. We define a Lie bracket multiplication on this set so that it becomes a Lie algebra. In the particular case of tangent bundles this Lie algebra coincides with the Lie algebra of vector fie...

Let X be a smooth variety de ned over an algebraically closed eld of arbitrary characteristic and OX H be a very ample line bundle on X We show that for a semistable X bundle E of rank two there exists an integer m depending only on E H X and H X such that the restric tion of E to a general divisor in jmH j is again semistable As corollaries we obtain boundedness results and weak versions of Bo...

Given a complex manifold X, a normal crossing divisor D ⊂ X whose irreducible components D 1 ,. .. , D s are smooth, and a choice of natural numbers r = (r 1 ,. .. , r s), we construct a manifold X(D, r) with an action of a torus Γ and we prove that some full subcategory of the category of Γ-equivariant vector bundles on X(D, r) is equivalent to the category of parabolic vector bundles on (X, D...