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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.01216 |
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| _version_ | 1866908289481048064 |
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| author | Xie, Junyi |
| author_facet | Xie, Junyi |
| contents | Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $τ: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every $i=0,\dots,2d$, the spectral radius of $f^*$ on the numerical group $N^i(X)\otimes \mathbb{R}$ and on the $l$-adic cohomology group $H^{2i}(X_{\overline{\mathbf{k}}},\mathbb{Q}_l)\otimes \mathbb{C}$ are the same. As a consequence, if $f$ is $q$-polarized for some $q>1$, we show that the norm of every eigenvalue of $f^*$ on the $j$-th cohomology group is $q^{j/2}$ for all $j=0,\dots, 2d.$ This generalizes Deligne's theorem for Weil's Riemann Hypothesis to arbitary polarized endomorphisms and proves a conjecture of Tate. We also get some applications for the counting of fixed points and its ``moving target" variant.
Indeed we studied the more general actions of certain cohomological coorespondences and we get the above results as consequences in the endomorphism setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical spectrums control Cohomological spectrums Xie, Junyi Algebraic Geometry Let $X$ be a smooth irreducible projective variety over a field $\mathbf{k}$ of dimension $d.$ Let $τ: \mathbb{Q}_l\to \mathbb{C}$ be any field embedding. Let $f: X\to X$ be a surjective endomorphism. We show that for every $i=0,\dots,2d$, the spectral radius of $f^*$ on the numerical group $N^i(X)\otimes \mathbb{R}$ and on the $l$-adic cohomology group $H^{2i}(X_{\overline{\mathbf{k}}},\mathbb{Q}_l)\otimes \mathbb{C}$ are the same. As a consequence, if $f$ is $q$-polarized for some $q>1$, we show that the norm of every eigenvalue of $f^*$ on the $j$-th cohomology group is $q^{j/2}$ for all $j=0,\dots, 2d.$ This generalizes Deligne's theorem for Weil's Riemann Hypothesis to arbitary polarized endomorphisms and proves a conjecture of Tate. We also get some applications for the counting of fixed points and its ``moving target" variant. Indeed we studied the more general actions of certain cohomological coorespondences and we get the above results as consequences in the endomorphism setting. |
| title | Numerical spectrums control Cohomological spectrums |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2412.01216 |