Generalized mu-ordinary Hasse invariants

Groupes $p$-divisibles avec condition de Pappas-Rapoport et invariants de Hasse

We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $μ$ ; this condition consists in a filtration on the sheaf of differentials $ω_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying …

Hilbert模形式中零 genus 曲线的部分Hassé不变量

Partial Hasse invariants for genus zero curves in Hilbert modular varieties

本文研究了Hilbert模形式中零 genus 曲线的部分Hassé不变量的特征零提升，并通过计算Teichmüller曲线得到类似Deuring的公式。

广义代数Morse不等式与Hasse-Schmidtjet微分

Generalized algebraic Morse inequalities and Hasse-Schmidt jet differentials

本文通过代数方法证明了Demailly定理，展示了一般型复射影流形上存在大量Green-Griffiths jet微分，并引入新的代数Morse不等式，适用于正特征情况，证明了光滑射影流形上存在Hasse-Schmidt jet微分。

Strata Hasse invariants, Hecke algebras and Galois representations

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …

Abelian varieties of prescribed order over finite fields

Given a prime power $q$ and $n \gg 1$, we prove that every integer in a large subinterval of the Hasse--Weil interval $[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$ is $#A(\mathbb{F}_q)$ for some geometrically simple ordinary principally polarized abelian variety $A$ of dimension $n$ over $\mathbb{F}_q$. As a consequence, we generalize a result of Howe and Kedlaya for $\mathbb{F}_2$ to show that for each prime power $q$, every sufficiently large positive integer is realizable, i.e., $#A(\mathbb{F}_q)$ for some abelian variety $A$ over $\mathbb{F}_q$. Our result also improves upon the best known const…

Zip 周期映射的 Ogus 原理：通过“Frobenius 与 Hodge 滤过”研究 Hasse 不变量的消失阶

An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'

本文将 Ogus 关于 Calabi–Yau 族 Hasse 不变量消失阶与共轭线位置的结果推广到群论 Zip 周期映射，并导出 Shimura 簇及 K3 模空间上的原理。

Serre-Tate theory for Calabi-Yau varieties

Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…

Generalized Serre--Tate Ordinary Theory

We study a generalization of Serre--Tate theory of ordinary abelian varieties and their deformation spaces. This generalization deals with abelian varieties equipped with additional structures. The additional structures can be not only an action of a semisimple algebra and a polarization, but more generally the data given by some ``crystalline Hodge cycles'' (a $p$-adic version of a Hodge cycle in the sense of motives). Compared to Serre--Tate ordinary theory, new phenomena appear in this generalized context. We give an application of this theory to the existence of ``good'' integral models of…

Zariski不变量用于拟普通超曲面

Zariski invariant for quasi-ordinary hypersurfaces

本文为拟普通参数化引入$\tilde{\mathcal{A}}$-不变量，用于描述具有一个广义特征指数且模空间可数的拟普通曲面。

On Generalized Wronskians

The Wroński determinant ({\em Wrońskian}), usually introduced in standard courses in Ordinary Differential Equations (ODE), is a very useful tool in algebraic geometry to detect ramification loci of linear systems. The present survey aims to describe some "materializations" of the Wrońskian and of its close relatives, {\it the generalized Wrońskians}, in algebraic geometry. Emphasis will be put on the relationships between Schubert Calculus and Linear Ordinary Differential Equations.}

Examples of authmorphism groups of ind-varieties of generalized flags

We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of maximal flags indexed by Z_{>0}. We pay special attention to differences with the case of ordinary flag varieties.

Quantum generalized cohomology

We construct a ring structure on complex cobordism tensored with the rationals, which is related to the usual ring structure as quantum cohomology is related to ordinary cohomology. The resulting object defines a generalized two- dimensional topological field theory taking values in a category of spectra.