An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917633831469056 |
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| author | Goldring, Wushi Reppen, Stefan |
| author_facet | Goldring, Wushi Reppen, Stefan |
| contents | This paper generalizes a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\text{dR}}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,μ,r)$ consisting of a connected, reductive $\mathbf{F}_p$-group $G$, a cocharacter $μ\in X_*(G)$ and an $\mathbf{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $ζ:X \to \text{$G$-$\mathtt{Zip}$}^μ$ is a smooth morphism (=`Zip period map'), then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_05707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration' Goldring, Wushi Reppen, Stefan Algebraic Geometry Number Theory This paper generalizes a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\text{dR}}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,μ,r)$ consisting of a connected, reductive $\mathbf{F}_p$-group $G$, a cocharacter $μ\in X_*(G)$ and an $\mathbf{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $ζ:X \to \text{$G$-$\mathtt{Zip}$}^μ$ is a smooth morphism (=`Zip period map'), then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces. |
| title | An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration' |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2404.05707 |