Divisibility of Frobenius eigenvalues on $\ell$-adic cohomology

v2: 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 $\ell$-adic cohomology have higher than known $q$-divisibility beyond the middle dimension. This sharpens both Deligne's integrality theorem and the cohomological divisibility theorem proven by the first author and N. Katz. Similar lower bounds are proved for the Hodge level for a complex variety beyond the middle dimension, improving earlier results in this direction. We discuss the affine case. The previous version contained a gap at this place. We are thankful to Dingxin Zhang for noticing it.

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