In this paper, we investigate the necessary sufficient conditions for the exactness of the homotopy sequence of Nori's fundamental group and apply these to various special situations to regain some classical theorems and give a counter example to show the conditions are not always satisfied. This work partially bases on the earlier work of H.Esnault, P.H.Hai, E.Viehweg.
In this article, we study the log-scheme theoretic version of the Nori fundamental group scheme. Similar to the classical Nori fundamental group scheme, the log Nori fundamental group scheme classifies torsors on log flat topology. We also compute the nilpotent pro-p quotient of the log Nori fundamental group scheme with log at $\{0, 1, \infty\}$.
In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic cur…
We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack $X$ over a field $k$. This is done by describing the Nori fundamental gerbe of an essentially finite cover of $X$. A similar result is also obtained for the $S$-fundamental gerbe.
In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field $k$. We give particular attention to the case of fields: to any field extension $K/k$ we attach a pro-local group scheme over $k$. We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of t…
Let $S$ be a connected Dedekind scheme and $X$ be a proper smooth connected scheme over $S$ . Let $D$ a divisor with no multiplicity of $X$ such that the irreducible components of $D$ and as well their intersections are smooth over $S$. Now if we endow $X$ with the log structure associated with $D$ then the structure morphism from $X$ to $S$ is log-smooth. Let $x: S \to X$ be a $S$-point such that it doesn't intersect $D$. Then we prove that the maximal abelian quotient of the log Nori fundamental group scheme of $X$ fits in to an exact sequence of the form $0 \rightarrow (\mathbf{NS}^τ_{X/S,D…
In this paper we describe the fundamental group-scheme of a proper variety fibered over an abelian variety with rationally connected fibers over an algebraically closed field. We use old and recent results for the Nori fundamental group-scheme, and of finite group-schemes in general to prove that the kernel of such a fibration is finite, and that the homotopy exact sequence holds in this case. As an application, we describe the fundamental group-schemes of certain proper varieties that are connected by curves or are related to them.
Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $A$ be an abelian surface over $k$. Fix an integer $n \geq 1$ such that $p \nmid n$ and let $K^{[n]}$ be the $n$-th Generalized Kummer Variety associated to $A$. In this article we aim to find the $S$-fundamental group scheme and Nori's fundamental group scheme of $K^{[n]}$.
Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.
Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $X$ be an irreducible smooth projective surface over $k$. Fix an integer $n \geq 1$ and let ${\mathcal{H}{\it ilb}}_X^n$ be the Hilbert scheme parameterizing effective $0$-cycles of length $n$ on $X$. The aim of the present article is to find the $S$-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme $\mathcal{H}{\it ilb}_X^n$.
In \cite{KP}, the last two authors introduced formal orbifold curves defined over an algebraically closed field of positive characteristics. They studied both étale and Nori fundamental group schemes associated to such objects. Our aim here is to study the higher dimensional analog of these objects objects and their fundamental groups.
Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar Conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's étale fundamental group of $U$. In this paper, we consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme, and give a partial answer to it.
Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_η$. Let $f: Y \rightarrow X_η$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_η$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $ψ^{(p')}: π(X_η,x_η)^{(…
We construct a log algebraic version of the homotopy sequence for a quasi-projective normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.
We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fulton's intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by Déglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Bar…
The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual étale fundamental group $π_1^{\mathrm{et}}$ defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes, generalizing finite étale covers. We prove exactness of the general homotopy sequence for the pro-étale fundamental group, i.e. that for a geometric point $\bar{s}$ on $S$ and a flat proper morphism $X \rightarrow S$ of finite presentation whose geometric fibres are connected and reduced, the sequence $$ π_1^{\mathrm{proet}}(X_{\bar{s}}) \rightarrow π_1^{\mathrm{proe…
Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences \begin{equation*} Π_{<\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to Π_{<\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \qquad \text{and} \qquad \widehatΠ{}_{\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to \widehatΠ{}_{\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \end{equation*} of protruncated and profinite étale homotopy types are…
The purpose of this work is to study the notion of bivariant theory introduced by Fulton and MacPherson in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism. We introduce several kinds of bivariant theory associated with a suitable ring spectrum and we construct a system of orientations (called fundamental classes) for global complete intersection morphisms between arbitrary schemes. This fundamental classes satisfies all the expected properties from classical intersection theory and lead to Gysin morphisms, Rieman…
本文的目的是将沙利文的有理同伦理论推广到非幂零空间，提供了一种替代方法来定义Toen的示意同伦类型在任何域k的特征零。新的功能包括一个明确的描述同伦群使用的Maurer-Cartan方程，收敛谱序列比较示意性的同伦群与上同调的普遍半简单的本地系统，和一个概括的Baues-Lemaire猜想。对于紧致Kaehler流形，其示意同伦群可以用这个上同调环来明确描述，给出了典型的权分解。还有极小模型、无点同伦类型和代数自同构群的概念。对于一个具有代数好的基本群和有限秩的高阶同伦群的空间，证明了其图解同伦群为π n（X）\otimes k.
