Cohomological divisibility and point count divisibility

Abstract

Let $X\subset {\mathbb P}^n$ be a closed scheme defined by r homogeneous equations of degrees $d_1\ge d_2\ge \dotsb \ge d_r$ over the finite field ${\mathbb F}_q$, with complement $U:={\mathbb P}^n\setminus X$. Let $\kappa$ be the maximum of 0 and the integral part of the rational number $({n-d_2-\dotsb-d_r})/{d_1}$. We show that the eigenvalues of the geometric Frobenius endomorphism acting on the $\ell$-adic cohomology $H^i_{\rm c}(U\times_{{\mathbb F}_q}\overline{{\mathbb F}_q}, {\mathbb Q}_\ell)$ with compact supports are divisible by $q^\kappa$ as algebraic integers.