## Algebraic Geometry Seminar

Seminar information archive ～08/05｜Next seminar｜Future seminars 08/06～

Date, time & place | Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu |

### 2024/07/04

13:00-14:30 Room #ハイブリッド開催/118 (Graduate School of Math. Sci. Bldg.)

On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)

**Stefan Reppen**(University of Tokyo)On a principle of Ogus: the Hasse invariant's order of vanishing and "Frobenius and the Hodge filtration'' (English)

[ Abstract ]

In joint work with W. Goldring we generalize 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 $H^n_{\dR}(Y/X)$ 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 $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, 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. In the talk I will present this work.

In joint work with W. Goldring we generalize 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 $H^n_{\dR}(Y/X)$ 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 $(G,\mu,r)$ consisting of a connected, reductive $\mathbb{F}_p$-group $G$, a cocharacter $\mu \in X_*(G)$ and an $\mathbb{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $\zeta:X \to \GZip^{\mu}$ is a smooth morphism, 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. In the talk I will present this work.