Let X be a closed equidimensional local complete intersection subscheme of a smooth projective scheme Y over a field, and let Xt denote the t-th thickening of X in Y. Fix an ample line bundle OY(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.
Bibliographical noteFunding Information:
B.B. was supported by NSF Grant DMS 1801689, a Packard fellowship, and the Simons Foundation Grant 622511, M.B. by DFG Grant SFB/TRR45, G.L. by NSF Grant DMS 1800355, A.K.S. by NSF Grant DMS 1801285, and W.Z. by NSF Grant DMS 1752081. The authors are grateful to the Institute for Advanced Study for support through the Summer Collaborators Program, to the American Institute of Mathematics for support through the SQuaREs Program, and to the referee for several helpful comments.
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