Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910871896195072 |
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| author | Hu, Fei Truong, Tuyen Trung Xie, Junyi |
| author_facet | Hu, Fei Truong, Tuyen Trung Xie, Junyi |
| contents | Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$.
Assuming the standard conjecture $D$, we prove
a weaker form of the Dynamical Degree Comparison conjecture;
equivalence of semisimplicity of Frobenius endomorphism and of any polarized endomorphism (a more general result, in terms of the biggest size of Jordan blocks, holds).
We illustrate these results through examples, including varieties dominated by rational maps from Abelian varieties and suitable products of $K3$ surfaces.
Using the same idea, we provide a new proof of the main result in a recent paper by the third author, including Tate's question/Serre's conjecture that for a polarized endomorphism $f:X\rightarrow X$, all eigenvalues of the action of $f$ on $H^k(X)$ have the same absolute value. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture Hu, Fei Truong, Tuyen Trung Xie, Junyi Algebraic Geometry Dynamical Systems Number Theory Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$. Assuming the standard conjecture $D$, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of semisimplicity of Frobenius endomorphism and of any polarized endomorphism (a more general result, in terms of the biggest size of Jordan blocks, holds). We illustrate these results through examples, including varieties dominated by rational maps from Abelian varieties and suitable products of $K3$ surfaces. Using the same idea, we provide a new proof of the main result in a recent paper by the third author, including Tate's question/Serre's conjecture that for a polarized endomorphism $f:X\rightarrow X$, all eigenvalues of the action of $f$ on $H^k(X)$ have the same absolute value. |
| title | Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture |
| topic | Algebraic Geometry Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2503.09432 |