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arXiv · 2511.17109

Parity and symmetry of polarized endomorphisms on cohomology

Fei Hu
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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.

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Fei Hu. 2025-11-21. Parity and symmetry of polarized endomorphisms on cohomology. https://arxiv.org/abs/2511.17109

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