Ample Vector Bundles on Open Algebraic Varieties

Let X be a smooth projective variety over the complex number field. Let D — X ,D | be an effective, reduced divisor on X with only simple normal crossings, DI denoting its irreducible components. A vector bundle E defined on X is said to be ample modulo D, if for every torsion free sheaf J^ on X9 there exists an integer m0>0 such that J^(x)S (E) \ X_D is generated by H°(X, J*(8)S (£)) for m>m 0 . A line bundle L is very ample modulo D, by definition, if the rational map 0\Li attached to the linear system |L| on X gives an embedding of X—D into some projective space. There are many question in which we come across line bundles which are not ample but ample modulo D for suitable D. For instans, let X be a toroidal compactification of a locally symmetric space of rank one of non-compact type and take D to be the boundary. In this case K x + D is nef and ample modulo D, but it is not ample in general (e. g. X is a ball-quotient surface with ellipitic curves as the cusps). The principle aim of the present article is to generalize a theorem of Demailly on ample line bundles (i. e. the case D=Q) '.