Invariance of the tame fundamental group under base change between algebraically closed fields
Aaron Landesman
本文证明连通正规有限型分离概形的驯顺 étale 基本群在特征 $p\geq0$ 的代数闭域间基变换下具有不变性。
我们证明，在特征 $p \geq 0$ 的代数闭域之间进行基变换时，连通、正规、有限型且分离的概形的驯顺 étale 基本群保持不变。
We show that the tame étale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic $p \geq 0$.
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Let X be a separated scheme of finite type over an algebraically closed field k and let m be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod m Suslin homology and the first mod m tame etale cohomology of X. We show that the induced homomorphism from the mod m Suslin homology to the abelianized tame fundamental group of X mod m is surjective. It is an isomorphism of finite abelian groups if (m, char(k)) = 1, and for general m if resolution of singularities holds over k.
Let $p$ be a prime number, and let $k$ be an algebraically closed field of characteristic $p$. We show that the tame fundamental group of a smooth affine curve over $k$ is a projective profinite group. We prove that the fundamental group of a smooth projective variety over $k$ is finitely presented. More generally we prove that the tame fundamental group of a smooth quasi-projective variety over $k$ which admits a good compactification is finitely presented. v2: references added. Thank you to all for the friendly and fruitful comments.
Homotopy invariance of tame homotopy groups of regular schemes
针对Artin-Mazur平展同伦群仅特征零具同伦不变性的问题，本文引入禁野分歧的驯拓扑，证明正则概形的驯同伦群满足$\text{A}^1$-同伦不变性。
The tame fundamental group schemes of curves in positive characteristic
本文研究正特征代数闭域上光滑曲线的驯服基本群概形，发展余特化映射理论，证明其依赖曲线且可重构数值不变量。
Anabelian properties of tame fundamental groups of singular curves
本文利用 Das 的定理和已知 anabelian 结果，在特定条件下确定了多种域上奇异曲线正规化的同构类，从而推进了奇异曲线 anabelian 性质的研究。
The tame Nori fundamental group
本研究针对代数闭域上的正规拟射影簇，引入Nori基本群概形的三种驯顺性概念，证明了在具备严格正规交叉除子补集的光滑完备化条件下三者等价，并建立了相应的Lefschetz型限制定理。
Tame fundamental groups of rigid spaces
该研究定义了刚性空间的驯服平展基本群，证明其在特定条件下拓扑有限生成或有限表现，为相关几何研究提供了关键有限性结论。
In this paper we prove that over algebraically closed field $K$ of positive characteristic $\neq 2$ every automorphism of the group of origin-preserving automorphisms of the polynomial algebra $K[x_1,\ldots, x_n]$ ($n>3$) which fixes every diagonal matrix preserves, up to composition with a linear inner automorphism, every tame automorphism.
Tame Parahoric Nonabelian Hodge Correspondence in Positive Characteristic over Algebraic Curves
本文证明了正特征代数曲线上驯顺 $G$-局部系统与对数 $G$-Higgs 丛间的非阿贝尔 Hodge 对应，并引入 parahoric 群概型语言完整刻画了非紧情形。
Let X be a smooth variety over an algebraically closed field k of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. O_X-coherent D_X-modules) on X without relying on resolution of singularities. The main result is that the category of regular singular stratified bundles with finite monodromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional k-vector spaces. As a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is…
On Brauer groups of tame stacks
本文通过比较温驯代数栈与其粗模空间的布劳尔群，发展计算栈的布劳尔群的通用工具，并应用于椭圆曲线模栈和代数闭域上的栈曲线。
On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]
本文研究了K[X,Y]局部有限导数的tame等距群，通过Van den Essen的分类确定各规范形式对应的tame等距群，并证明其与关联的指数自同构的tame等距群一致，与全自同构群行为不同。
We study rational points on a smooth variety X over a complete local field K with algebraically closed residue field, and models of X with tame quotient singularities. If a model of X is the quotient of a Galois action on a weak Néron model of the base change of X to a tame Galois extension of K, then we construct a canonical weak Néron model of X with a map to this model, and examine its special fiber. As an application we get examples of singular models of X such that X has K-rational points specializing to a singular point of this model. Moreover we obtain formulas for the motivic Serre inv…
For every adic space $Z$ we construct a site $Z_t$, the tame site of $Z$. For a scheme $X$ over a base scheme $S$ we obtain a tame site by associating with $X/S$ an adic space $\textit{Spa}(X,S)$ and considering the tame site $\textit{Spa}(X,S)_t$. We examine the connection of the cohomology of the tame site with étale cohomology and compare its fundamental group with the conventional tame fundamental group. Finally, assuming resolution of singularities, for a regular scheme $X$ over a base scheme $S$ of characteristic $p > 0$ we prove a cohomological purity theorem for the constant sheaf $\ma…
Let $X$ and $Y$ be nonsingular projective varieties over an algebraically closed field $k$ of positive characteristic. If $X$ and $Y$ are birational, we show their $S$-fundamental group schemes are isomorphic.
In this paper, we show that general Fano complete intersections over an algebraically closed field of arbitrary characteristics are separably rationally connected. Our proof also implies that general log Fano complete intersections with smooth tame boundary divisors admit very free $A^1$-curves.
Let $X$ be a generic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p\geq 0$. We show that for $n$ sufficiently large there exists a tame rational map $f:X\to \PP^1_k$ with monodromy group $A_n$. This generalizes a result of Magaard--Völklein to positive characteristic.
Unirational algebraic groups and tame ramification
研究无理代数群在基变换成整环时Néron lft模型的行为，揭示跳跃现象与动机ζ函数的有理性，回答Halle-Nicaise和Edixhoven的问题。
The Milnor Number of Plane Branches With Tame Semigroup of Values
本文研究任意特征代数闭域上平面分支的Milnor数，证明当值半群为驯顺半群（域特征不整除其极小生成元）时，Milnor数仍与半群导子相等，弥补了正特征域下原结论不成立的缺口。
We discuss Suslin's singular homology and cohomology. In the first half we examine the p-part in characteristic p, and the situation over non-algebraically closed fields. In the second half we focus on finite base fields. We study finite generation properties, and give a modified definition which behaves like a homology theory: in degree zero it is a copy of Z for each connected component, in degree one it is related to the abelianized (tame) fundamental group, even for singular schemes, and it is expected to be finitely generated in general.
On the local étale fundamental group of KLT threefold singularities
本文证明了正特征$p>5$代数闭域上KLT三维奇点的局部étale基本群是驯顺且有限的，并建立了大开子集上有限幂零torsor的提升性质。
Fundamental group schemes of globally F-regular varieties
该研究证明正特征代数闭域上连通正规整体F-正则射影簇的连通拟挠子覆盖塔稳定，其正则轨迹的Nori基本群概型有限且线性约化，有限性是Xu和Braun相关定理的正特征类比。
We compute the étale cohomology groups H^i(X,G_m) in several cases, where X is a smooth tame Deligne-Mumford stack of dimension 1 over an algebraically closed field. We have complete results for orbicurves (and, more generally, for twisted nodal curves) and in the case all stabilizers are cyclic; we give some partial results and examples in the general case. In particular we show that if the stabilizers are abelian then H^2(X, G_m) does not depend on X but only on the underlying orbicurve and on the generic stabilizer.
Local types of $(Γ,G)$-bundles and parahoric group schemes
本文利用弱分歧点处稳定子群的第一非阿贝尔群上同调分类计算了光滑射影曲线上 $(Γ,G)$-丛的局部类型，并证明了特征零下泛单连通 parahoric Bruhat--Tits 群概型可由该丛导出及其局部版本。
Galois Covers of Singular Curves in Positive Characteristics
本文研究正特征域上奇异曲线的平展基本群结构，证明射影和仿射情形下其可分解为正规化基本群与自由投射有限群的自由积，并给出有限群作为伽罗瓦群出现的完整分类。
This paper is an attempt to give some general results on the tempered fundamental group of a $p$-adic smooth algebraic varieties (which is a sort of analog of the topologic fundamental group of complex algebraic varieties in the p-adic world). The main result asserts that one can recover the metric structure of the graph of the stable model of a Mumford curve from the tempered fundamental group of this Mumford curve. We will also prove birational invariance, invariance by algebraically closed extensions and a Kuenneth formula for the tempered fundamental group. We will describe the tempered fu…
If $C$ is a smooth curve over an algebraically closed field $k$ of characteristic $p$, then the structure of the maximal prime to $p$ quotient of the étale fundamental group is known by analytic methods. In this paper, we discuss the properties of the fundamental group that can be deduced by purely algebraic techniques. We describe a general reduction from an arbitrary curve to the projective line minus three points, and show what can be proven unconditionally about the maximal pro-nilpotent and pro-solvable quotients of the prime-to-$p$ fundamental group. Included is an appendix which treats …
On a theorem of Narasimhan and Ramanan on deformations
本文基于模栈理论，研究了稳定扭量的上同调群，并推导了Narasimhan和Ramanan的经典结果。
Let $K$ be a field. The étale open topology on the $K$-points $V(K)$ of a $K$-variety $V$ was introduced in our previous work. The étale open topology is non-discrete if and only if $K$ is large. If $K$ is separably, real, $p$-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. We show that existentially definable sets in perfect large fields behave well with respect to this topology: such sets are finite unions of étale open subsets of Zariski closed sets. This implies that existentially definable sets in arbitrary perfect large field…
Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In "Flat vector bundles and the fundamental group in non-zero characteristics" (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975)), Gieseker conjectured that every stratified bundle (i.e. every O-coherent D-module) on X is trivial, if and only if the étale fundamental group of X is trivial. This was proven by Esnault-Mehta in "Simply connected projective manifolds in characteristic p> 0 have no nontrivial stratified bundles" (Invent. Math. 181 (2010)). Building on the classical situa…
We show, for a smooth projective variety $X$ over an algebraically closed field $k$ with an effective Cartier divisor $D$, that the torsion subgroup $\CH_0(X|D)\{l\}$ can be described in terms of a relative {é}tale cohomology for any prime $l \neq p = {\rm char}(k)$. This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including $p$-torsion) for $\CH_0(X|D)$ when $D$ is reduced. We deduce applications to the problem of invariance of the prime-to-$p$ torsion in $\CH_0(X|D)$ under an infinitesimal exten…
On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces
本文研究了有限群作用下奇异曲线和紧复空间的上同调表示，推导了Chevalley--Weil型公式，并探讨了节点曲线的不变截面和变形空间。
We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $Γ_\infty$, where $Γ$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $Γ_\infty$. In both contexts, we show that the corresponding cohomology theory satisf…
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