Variation of Tannaka groups of perverse sheaves in family
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908472316002304 |
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| author | Cadoret, Anna Liu, Haohao |
| author_facet | Cadoret, Anna Liu, Haohao |
| contents | Let $k$ be a field of characteristic $0$, let $S$ be a smooth, geometrically connected variety over $k$, with generic point $η$, and $f:\mathbb{X}\rightarrow S$ a morphism separated and of finite type. Fix a prime $\ell$. Let $\mathbb{P}$ be an $f$-universally locally acyclic relative perverse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbb{X}/S$. We prove that if for some (equivalently, every) geometric point $\bar η$ over $η$ the restriction $\mathbb{P}|_{\mathbb{X}_{\bar η}}$ is simple as a perverse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbb{X}_{\bar η}$, then there is a non-empty open subscheme $U\subset S$ such that, for every geometric point $\bar s$ on $U$, the restriction $\mathbb{P}|_{\mathbb{X}_{\bar s}}$ is simple as a perverse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbb{X}_{\bar s}$. When $f:\mathbb{X}\rightarrow S$ is an abelian scheme, we give applications of this result to the variation with $s\in S$ of the Tannaka group of $\mathbb{P}|_{\mathbb{X}_{\bar s}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variation of Tannaka groups of perverse sheaves in family Cadoret, Anna Liu, Haohao Algebraic Geometry Number Theory Let $k$ be a field of characteristic $0$, let $S$ be a smooth, geometrically connected variety over $k$, with generic point $η$, and $f:\mathbb{X}\rightarrow S$ a morphism separated and of finite type. Fix a prime $\ell$. Let $\mathbb{P}$ be an $f$-universally locally acyclic relative perverse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbb{X}/S$. We prove that if for some (equivalently, every) geometric point $\bar η$ over $η$ the restriction $\mathbb{P}|_{\mathbb{X}_{\bar η}}$ is simple as a perverse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbb{X}_{\bar η}$, then there is a non-empty open subscheme $U\subset S$ such that, for every geometric point $\bar s$ on $U$, the restriction $\mathbb{P}|_{\mathbb{X}_{\bar s}}$ is simple as a perverse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbb{X}_{\bar s}$. When $f:\mathbb{X}\rightarrow S$ is an abelian scheme, we give applications of this result to the variation with $s\in S$ of the Tannaka group of $\mathbb{P}|_{\mathbb{X}_{\bar s}}$. |
| title | Variation of Tannaka groups of perverse sheaves in family |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2505.01716 |