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 $$\mathcal {O}_Y(1)$$ on Y. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer c, such that for all integers  $$t \geqslant 1$$ , the cohomology group $$H^k(X_t,\mathcal {O}_{X_t}(j))$$ 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.