Parity and symmetry of polarized endomorphisms on cohomology

We show that the eigenvalues of any polarized endomorphism acting on the $\mathrm{\ell }$-adic étale cohomology of a smooth projective variety satisfy certain parity and symmetry properties, as predicted by the standard conjectures. These properties were previously known for Frobenius endomorphisms. Besides the hard Lefschetz theorem, a key new ingredient is a recent Weil’s Riemann hypothesis-type result due to J.~Xie. We also prove a “Newton over Hodge” type property for abelian varieties and Grassmannians.