2020/02/03 by Bhatt, Bhargav, Blickle, Manuel, Lyubeznik, Gennady +2
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2002.00913
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 \geqslant 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.