AN EXTENSION OF THE VANISHING THEOREM OF KAWAMATA AND VIEHWEG TO NEF VECTOR-BUNDLES OF ARBITRARY RANK OVER SMOOTH PROJECTIVE VARIETIES

The vanishing theorem of Kawamata and Viehweg is an important extensio n of Kodaira's theorem to nef and big line bundles. We extend it to ne f vector bundles of arbitrary rank over smooth projective varieties : the hypothesis of a positive self intersection becomes a positivity co ndition for caracteristic numbers defined by certain Schur polynomials . We derive this condition from an expression of the self-intersection of a line bundle on a relative flag manifold, which provides some ins ight into the corresponding Gysin morphism. This expression is itself a byproduct of some expansions of the Chem character of symmetric powe rs, that should be of independant interest.