Parity and symmetry of polarized endomorphisms on cohomology
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909916293234688 |
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| author | Hu, Fei |
| author_facet | Hu, Fei |
| contents | We show that the eigenvalues of any polarized endomorphism acting on the $\ell$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parity and symmetry of polarized endomorphisms on cohomology Hu, Fei Algebraic Geometry Number Theory 14A10, 14G17, 14F20, 14F40, 15A63 We show that the eigenvalues of any polarized endomorphism acting on the $\ell$-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. |
| title | Parity and symmetry of polarized endomorphisms on cohomology |
| topic | Algebraic Geometry Number Theory 14A10, 14G17, 14F20, 14F40, 15A63 |
| url | https://arxiv.org/abs/2511.17109 |