Divisibility of Frobenius eigenvalues on l-adic cohomology

For a projective variety defined over a finite field with q elements, it is shown that as algebraic integers, the eigenvalues of the geometric Frobenius acting on l-adic cohomology have higher than known q-divisibility beyond the middle dimension. This sharpens both Deligne's integrality theorem [2, Corollary 5.5.3] and the cohomological divisibility theorem [10, Theorem 4.1]. A similar lower bound is proved for the Hodge level for a complex projective variety beyond the middle dimension, improving earlier results in this direction. The affine version of our results for the compactly supported cohomology is still open in general.