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 the tame fundamental group from a stack-theoretic perspective.
We introduce a spreading out technique to deduce finiteness results for étale fundamental groups of complex varieties by characteristic $p$ methods, and apply this to recover a finiteness result proven recently for local fundamental groups in characteristic $0$ using birational geometry.
Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of the pro-étale fundamental group of the closed fibre and the category of stratified bundles on the geometric generic fibre. By Tannakian duality, this functor induces a morphism between the corresponding affine group schemes. We show that this morphism is a lifting of the specialization map, constructed by Grothendieck, between the étale fundamental groups.
Let $X$ be a normal noetherian scheme and $Z \subseteq X$ a closed subset of codimension $\geq 2$. We consider here the local obstructions to the map $\hatπ_{1}(X\backslash Z) \to \hatπ_{1}(X)$ being an isomorphism. Assuming $X$ has a regular alteration, we prove the equivalence of the obstructions being finite and the existence of a Galois quasi-étale cover of $X$, where the corresponding map on fundamental groups is an isomorphism.
We define the fundamental group underlying to Lichtenbaum's Weil-étale cohomology for number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced in \cite{Lichtenbaum}. We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition given in \cite{Lichtenbaum-finite-field}. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents fi…
We follow the pattern in a recent paper of Otabe [Ota15] to define an action of the étale fundamental group scheme $π^\text{et}(X)$ on the local component of the essentially finite fundamental group scheme $π^{\mathrm{EF}}(X)$ of Nori. We show that the associated representation is faithful when $X$ is a curve of genus $\geq 2$.
We prove that a strongly $F$-regular scheme $X$ admits a finite, generically Galois, and étale-in-codimension-one cover $\widetilde X \to X$ such that the étale fundamental groups of $\widetilde X$ and $\widetilde X_{reg}$ agree. Equivalently, every finite étale cover of $\widetilde X_{reg}$ extends to a finite étale cover of $\widetilde X$. This is analogous to a result for complex klt varieties by Greb, Kebekus and Peternell.
We prove that the geometric etale fundamental group of a (geometrically connected) rigid smooth $p$-adic affinoid curve is a semi-direct factor of a certain profinite free group. We also prove that the maximal pro-$p$ (resp. maximal prime-to-$p$) quotient of this geometric étale fundamental group is pro-$p$ free of infinite rank (resp. (pro-)prime-to-$p$ free of finite computable rank).
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…
We introduce the category of finite étale covers of an arbitrary schematic finite space $X$ and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the étale fundamental group of schematic spaces. If $X$ is a finite model of a scheme $S$, we show that the resulting Galois theory on $X$ coincides with the classical theory of finite étale covers on $S$ and therefore we recover the classical étale fundamental group introduced by Grothendieck. In order to prove these results it is crucial to find a suitable geometric notion of connectedn…
We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.
In this paper we will give a computation of the étale fundamental group of an integral arithmetic scheme. For such a scheme, we will prove that the étale fundamental group is naturally isomorphic to the Galois group of the maximal formally unramified extension over the function field. It consists of the main theorem of the paper. Here, formally unramified will be proved to be arithmetically unramified which is defined in an evident manner and coincides with that in algebraic number theory. Hence, formally unramified has an arithmetic sense. At the same time, such a computation coincides with t…
Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. Using this we construct a specialization functor from the category of continuous representations of the pro-étale fundamental group of the special fiber to the category of $F$-divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the étale fundamental groups. We also express the pro-étale fundamental group of a connected scheme $X$ of finite type over…
The previous version of this paper relied on a paper by another author whose proof appears to be invalid in a fundamental way. In arXiv:1707.00649 the author, together with Jeff Yelton, came up with a new proof of almost identical results. (See Remark 1.4 of arXiv:1707.00649.)
Let X be a smooth or proper variety defined over a finite field. The geometric etale fundamental group of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Q_l-algebraic completion of the fundamental group as a non-abelian Weil representation. Lafforgue's Theorem and Deligne's Weil II theorems imply that this affine group scheme is mixed, in the sense that its structure sheaf is a mixed Weil representation. When X is smooth, weight restrictions apply, affecting the possibilities for the structure of this group. This gives new examples of group…
Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the étale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of étale …
Let $k$ be an algebraically closed field. Chambert-Loir proved that the étale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain connected variety over $k$ is finite and étale. In particular, the fundamental group scheme of a Fano variety is finite and étale.
For a formal scheme $\mathfrak{X}$ of finite type over a complete rank one valuation ring, we construct a specialization morphism \[ π^{\rm dJ}_1(\mathfrak{X}_η) \to π^{\rm proet}_1(\mathfrak{X}_k) \] from the de Jong fundamental group of the rigid generic fiber to the Bhatt-Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak{X}$ and normalizations of the irreducible components of $\mathfrak{X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, …
We call a projective surface $X$ mixed quasi-étale quotient if there exists a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C \rightarrow X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-étale surface. We study the mixed quasi-étale surfaces under the assumption that $(C\times C)/G^0$ has only nodes as singularities, where $G^0\triangleleft G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal reg…
This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that all irreducible stratified bundles have rank 1 if and only if the commutator $[π_1, π_1]$ of the étale fundamental group $π_1$ is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1 $ is abelian without $p$-power quotient. This answers positively a conject…
Let G be a connected algebraic group over an algebraically closed field of characteristic p (possibly 0), and X a variety on which G acts transitively with connected stabilizers. We show that any étale Galois cover of X of degree prime to p is also homogeneous, and that the maximal prime-to-p quotient of the étale fundamental group of X is commutative. We moreover obtain an explicit bound for the number of topological generators of the said quotient. When G is commutative, we also obtain a description of the prime-to-p torsion in the Brauer group of G.
Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $…
v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[π_1^{{\rm \acute{e}t}}, π_1^{{\rm \acute{e}t}}]$ of the étale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.
Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field k which is finitely generated over Q has a fundamental system of (affine) anabelian Zariski-neighbourhoods. This was predicted by Grothendieck in his letter to Faltings.
Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms $σ_i:x_i\to \mbox{F}^*x_{i+1}$. Let $\mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $\mathscr{X}\to Π_1(\mathscr{X}/S)$ the étale fundamental pro-groupoid, constructed in the present text. We prove that if $\mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $\mathscr{X}\to S$ has geometrically reduced fibres, there is a bifunctorial isomor…
Let GO(2n) be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic not equal to 2. We determine the etale cohomology ring with mod 2 coefficients of the algebraic stack BGO(2n). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring of the classifying space BGO(2n) of the complex Lie group GO(2n) in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are (i) an extension to etale cohomology of an idea of Totaro, originally used …
Working with a singular variety X, one is often interested in comparing the set of etale covers of X with that of its smooth locus X_reg. More precisely, one may ask: What are the obstructions to extend finite etale covers of X_reg to all of X? How do the etale fundamental groups of X and of its smooth locus compare? For projective varieties with Kawamata log terminal (klt) singularities we answer these questions in part. The main result of this paper asserts that there are no infinite towers of finite morphisms over a klt base variety where all morphisms are etale in codimension one, but bran…
Passing from arithmetic schemes to algebraic schemes, in a similar manner we will have the computation of the étale fundamental group of an algebraic scheme and then will define and discuss the qc fundamental group of an algebraic scheme in this paper. The qc fundamental group will also give a prior estimate of the étale fundamental group.
Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to sign, the leading term of the zeta-function $ζ_X(s)$ at $s=0$. In this paper we consider the category of sheaves $\bar{X}_L$ on this conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We show that $\bar{X}_L$ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topologic…
Let S be a minimal complex surface of general type with $q(S)=0$. We prove the following statements concerning the algebraic fundamental group: I) Assume that K^2_S\leq 3χ(S). Then S has an irregular etale cover if and only if S has a free pencil of hyperelliptic curves of genus 3 with at least 4 double fibres. II) If K^2_S=3 and χ(S)=1, then S has no irregular etale cover. III) If K^2_S<3χ(S) and S does not have any irregular etale cover, then the order of the algebraic fundamental group is lesser or equal to 9, and if equality occurs then K^2_S=2, χ(S)=1.
Let $K$ be the fraction field of a strictly Henselian DVR of characteristic $p \geq 0$ with algebraic closure $\bar{K}$, and let $α_{1}, ..., α_{d} \in \mathbb{P}_{K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$p$ étale fundamental group of $\mathbb{P}_K^1\smallsetminus\{α_1,...,α_d\}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $a_{1}, ..., a_{d}$ be distinct power series in $\mathbb{C}[[x]]$ with the same intersection behavior as the $α_i$'s, con…
Let k be a field, and let π:\tilde{X} -> X be a proper birational morphism of irreducible k-varieties, where \tilde{X} is smooth and X has at worst quotient singularities. When the characteristic of k is zero, a theorem of Kollár in [Ko1] says that π induces an isomorphism of etale fundamental groups. We give a proof of this result which works for all characteristics. As an application, we prove that for a smooth projective irreducible surface X over an algebraically closed field k, the etale fundamental group of the Hilbert scheme of n points of X, where n > 1, is canonically isomorphic to th…
We prove that the local etale fundamental group of a strongly $F$-regular singularity is finite (and likewise for the étale fundamental group of the complement of a codimension $\geq 2$ set), analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In fact our result is effective, we show that the reciprocal of the $F$-signature of the singularity gives a bound on the size of this fundamental group. To prove these results and their corollaries, we develop new transformation rules for the $F$-signature under finite etale-in-codimension-one extensions. …
The tempered fundamental group of a p-adic analytic variety classifies étale coverings that become topological coverings (for the Berkovich topology) after finite étale base change. I study what can be recovered from the tempered fundamental group. In particular, I prove that, for a Mumford curve, one can recover the metric of the graph of the stable reduction from the tempered fundamental group. I also study how the tempered fundamental group behaves along the fibers of a fibration.
We relate a part of the abelian etale fundamental group of curves over local fields to the component group of the Neron model of the jacobian. We apply the result to the modular curve X_0(p)/Q_p to show that the unramified abelian covering X_1(p) over X_0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X_0(p).
Formal orbifolds are defined in higher dimension. Their étale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to approximate the étale fundamental groups of normal varieties. Etale site on formal orbifolds are also defined. This framework allows one to study wild ramification in an organised way. Brylinski-Kato filtration, Lefschetz theorem for fundamental groups and $l$-adic sheaves in these contexts are also studied.
In this paper we will define a qc fundamental group for an arithmetic scheme by quasi-galois closed covers. Then we will give a computation for such a group and will prove that the etale fundamental group of an arithmetic scheme is a normal subgroup in our qc fundamental group, which make up the main theorem of the paper. Hence, our group gives us a prior estimate of the etale fundamental group. The quotient group reflects the topological properties of the scheme.
This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of étale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over ver…
We introduce an etale fundamental group with modulus and construct a reciprocity homomorphism from the Kato-Saito idele class group with modulus to this fundamental group. This is the K-theoretic analogue of the reciprocity for the cycle-theoretic idele class group with modulus due to Kerz-Saito, and plays a central role in showing the isomorphism between the two idele class groups. It also provides a new interpretation of the already known etale fundamental group with modulus due to Deligne and Laumon.
In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose complements are homeomorphic, but whose embeddings in $\mathbb{P}^2$ are not. Most of the known invariants used to detect Zariski pairs depend on the étale fundamental group. In the case of Galois-conjugate curves, their étale fundamental groups coincide. Braid monodromy factorizat…
Let $GO(2n)$ be the general orthogonal group scheme (the group of orthogonal similitudes). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$ of the classifying space $BGO(2n,\mathbb C)$ of the corresponding complex Lie group $GO(2n,\mathbb C)$ in terms of explicit generators and relations. The author of the present note showed that over any algebraically closed field of characteristic not equal to $2$, the smooth-étale cohomology ring $H_{\rm sm-ét}^*(BGO(2n),\mathbb F_2)$ of the classifying algebraic st…
