Stability and Cohomology of Kernel Bundles on Pn

Abstract

In this paper, we study the cohomology of vector bundles on ${\mathbb{P}}^{\mathit{n}}$ defined as kernels or cokernels of general maps ${\mathit{V}}_1\to {\mathit{V}}_2$, where ${\mathit{V}}_{\mathit{i}}$ 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 ${\mathbb{P}}^{\mathit{n}}$ and give a criterion for such bundles to be ample.