Parity and symmetry of polarized endomorphisms on cohomology

TL;DR

This paper proves that eigenvalues of polarized endomorphisms on the cohomology of smooth projective varieties exhibit specific symmetry and parity properties, extending known results for Frobenius and utilizing recent advances like Xie's Riemann hypothesis-type theorem.

Contribution

It establishes new symmetry and parity properties of eigenvalues for polarized endomorphisms, generalizing previous results and incorporating recent theoretical developments.

Findings

Eigenvalues satisfy symmetry and parity properties as predicted by conjectures.

Proved a 'Newton over Hodge' property for abelian varieties and Grassmannians.

Extended known results from Frobenius to more general polarized endomorphisms.

Abstract

We show that the eigenvalues of any polarized endomorphism acting on the $\ell$-adic \'etale 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.