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 $t⩾1$, 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.