An asymptotic vanishing theorem for the cohomology of thickenings

Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let X_t denote the t-th thickening of X in Y. Fix an ample line bundle O_Y(1) on Y. 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 Hk(Xt,OXt(j)) vanishes for k< dim X and j< - ct. Note that there are no restrictions on the characteristic of the field, or on 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.