Stability and Cohomology of Kernel Bundles on Pn

. In this paper, we study the cohomology of vector bundles on Pn defined as kernels or cokernels of general maps V1 -> V2, where Vi are direct sums of line bundles or certain exceptional bundles. This problem is deeply connected with fundamental results on the ranks of multiplication maps between spaces of homogeneous polynomials. Our main result establishes an asymptotic cohomology vanishing theorem. We also characterize the stability of general Steiner bundles on Pn and give a criterion for such bundles to be ample.