An Application of a Log Version of the Kodaira Vanishing Theorem to Embedded Projective Varieties

Abstract

Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain twists of powers of the ideal sheaf of Y. This is applied to get rather non-trivial vanishing in case Y is a universal determinantal variety (i.e. quadratic Veronese, Segre embedding or Grassmannian of P^1's) or a curve embedded by a complete linear series of high degree. The key in the determinantal cases is to use the theory of complete objects. While the theories of complete quadrics and linear maps yield the desired results, a corresponding theory of complete skew forms seems not have been previously worked out. The desired results are obtained in an appendix.