Th\'{e}or\`{e}mes d'annulation et lieux de d\'{e}g\'{e}n\'{e}rescence en petit corang

After giving an explicit description of all the non vanishing Dolbeault cohomology groups of ample line bundles on grassmannians, I give two series of vanishing theorems for ample vector bundles on a smooth projective variety. They imply a part of a conjecture by Fulton and Lazarsfeld about the connectivity of some degeneracy loci.