An asymptotic vanishing theorem for the cohomology of thickenings

Abstract

Let X be a closed local complete intersection subscheme of projective space over a field, and let X t denote its t-th thickening. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers t 1, the cohomology group H k (X t ,O Xt ( j)) vanishes for k < dim X and j < −ct. Note that there are no restrictions on the characteristic of the field or the singular locus of X. We also construct examples illustrating that a linear bound is indeed the best possible, and that the constant c is unbounded, even in a fixed dimension.