Richard Hain, Makoto Matsumoto
This should be the final version of this paper. Numberous minor improvements have been made to the manuscript, one argument has been corrected, and an appendix has been added.
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We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.
Let T be a Tannakian category over a field k of characteristic 0 and π(T) its fundamental group. In this paper we prove that there is a bijection between the otimes-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T-schemes of π(T). We apply this result to the Tannakian category T_1(k) generated by motives of niveau \leq 1 defined over k, whose fundamental group is called the motivic Galois group G_mot(T_1(k)) of motives of niveau \leq 1. We find four short exact sequences of affine group sub-T_1(k)-schemes of G_mot(T_1(k)), correlated one to each other by in…
In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our previous main results. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP on the category of noncommutative Chow motives. This allows us to introduce the correct non…
In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero, various unconditional approaches were later proposed. The most precise one, due to J. Ayoub, relies on Voevodsky theory of mixed motives and on a new tannakian theory. It sheds new light on periods of algebraic varieties, and shows in particular that polynomial relations between periods of a pencil of algebraic varieties always arise from Stokes formula.
Tannakian fundamental groups of blended extensions
该文研究特征零域上中性塔纳基范畴中混合扩张的塔纳基基本群的平凡作用子群，并将其应用于德利涅1-动机的霍奇猜想动机版本的幂幺部分问题。
We establish a tannakian formalism of $p$-adic multiple polylogarithms and $p$-adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de Rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its Hodge and etale analogues. As an application we give a way to erase log poles of $p$-adic multiple polylogarithms and introduce overconvergent $p$-adic multiple polylogarithms which might be $p$-adic multiple analogue of Zagier's single-valued complex polylogarithms.
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…
Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$ dominating $f$ by an $X$-morphism $λ:\hat{X}_Y\to Y$ and universal for this property. Moreover we show that $λ:\hat{X}_Y\to Y$ is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over $Y$ is still an essentially finit…
Tannakian duality and Gauss-Manin connections for a family of curves
本文研究了光滑曲线族的微分基本群方案与高斯-曼金连接之间的关系，证明了在高斯-曼金连接中几何相对基本群的上同调映射是同构的，从而将高斯-曼金连接解释为微分基本群的上同调。
We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R …
We study affine group schemes over a discrete valuation ring $R$ using two techniques: Neron blowups and Tannakian categories. We employ the theory developed to define and study differential Galois groups of $\mathcal D$-modules on a scheme over a $R$. This throws light on how differential Galois groups of families degenerate.
In the 1st part of this work [DHdS18], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of D-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of "infinite type", Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of D-modules. To conclude, we isolate a Tannakian property of affine group schemes, n…
The Hodge Galois group is the Tannakian Galois group of the category of real mixed Hodge structures. It has a subgroup, called the twistor Galois group, which is the Galois group of the category of mixed twistor structures, defined by C. Simpson. It is isomorphic to a semidirect product of C* and the unipotent radical of the Hodge Galois group. We define a natural action of the twistor Galois group by A-infinity autoequivalences of the derived category of complexes of sheaves with smooth cohomology on a compact smooth Kahler manifold X. The action of C* is provided by Simpson's action on irred…
In this paper we extend the generalized algebraic fundamental group constructed by Esnault and Hogadi to general fibered categories using the language of gerbes. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined by Borne and Vistoli for non smooth non pseudo-proper algebraic stacks.
Galois coinvariants and fundamental groups of arithmetic schemes
该研究针对有限生成域上的几何平展基本群与平展上同调，建立特定伽罗瓦扩张下共变式的有限性性质，进而得到数域上算术概型平展基本群的有限性结果。
Universality of the Galois action on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$
本文证明数域 Galois 群来自几何的任意半单表示都能实现为三穿孔射影直线基本群 pro-代数完备的正则函数环的子商，揭示了该基本群上 Galois 作用的普适性。
This paper establishes restrictions on the possible Galois actions on the pro-l-unipotent fundamental group of a smooth variety X of good reduction over a local field K. In particular, if X is proper and l is not equal to the residue characteristic p of K, then this action is uniquely determined by the action on cohomology. A similar result for l=p is then proved via a comparison theorem between the crystalline and pro-p homotopy types.
Let $\mathbf{T}$ be a neutral Tannakian category over a field of characteristic zero with unit object $\mathbf{1}$, and equipped with a filtration $W_\cdot$ similar to the weight filtration on mixed motives. Let $M$ be an object of $\mathbf{T}$, and $\underline{\mathfrak{u}}(M)\subset W_{-1}\underline{Hom}(M,M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of $Gr^WM$. A result of Deligne gives a characterization of $\underline{\mathfrak{u}}(M)$ in terms of the extensions $0\longrightarrow W_pM \longrightarrow M \longrightarr…
Suppose that C is a smooth, projective, geometrically connected curve of genus g > 2 defined over a number field K. Suppose that x is a K-rational point of C. Denote the Lie algebra of the unipotent completion (over Q_ell) of the fundamental group of the corresponding complex analytic curve by p(C,x). This is acted on by both G_K (the Galois group of K) and Gamma, the mapping class group of the pointed complex curve. In this paper we show that the algebraic cycle C_x-C_x^- in the jacobian of C controls the size of the image of G_K in Aut p(C,x). More precisely, we give necessary and sufficient…
Galois descent for motivic theories
本文给出了纤维化加性范畴成为stack的条件，适用于多种motive理论，并简化证明了motivic Galois群序列的正合性及Chow-Lefschetz motives的Galois群计算。
We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain `` motivic Galois group'', which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of th…
Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an étale Galois cover $Y\to X$ with group $N_G(P)$, then $G$ is the Galois group wan étale Galois cover $\mathcal{Y}\to\mathcal{X}$, where the genus of $\mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $…
Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the étale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundament…
This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number $ł$. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-$ł$ completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted …
We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).
We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $σ\in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^σ$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting p…
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.
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.
Étale Fundamental Groups: a geometric and topological approach to fundamental groups in algebraic geometry
本文通过拓扑覆盖空间、Étale拓扑和Tannakian对偶三种框架，系统构建并比较了代数几何中的基本群概念。
This is an essay to accompany the author's lecture at the introductory workshop on `Nonabelian fundamental groups in arithmetic geometry' at the Newton Institute, Cambridge in July, 2009.
Notes On Fundamental Groupoid Schemes
本文研究了对应于特定向量丛Tannakian范畴的基本群胚概形，计算了各向异性圆锥、Klein瓶和阿贝尔簇的基本群胚概形，并通过表示探讨了它们之间的关系。
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
Depth one part of Tannakian groups of filtrations
本文研究了在零特征下，Tannakian范畴中滤波对象的Tannakian基本群的Lie代数的深度一部分，通过扩展来描述其梯度部分，并探讨了混合动机及其实现在此一般情况下的应用。
A remark on Ext groups for motives with maximal unipotent radicals
本文研究具有极大幂单根的动机的Tannakian子范畴中的Ext^1群，并应用于1-动机和混合Tate动机，推导出Grothendieck周期猜想的若干推论。
We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito for the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We also compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.
This is the final paper in a series of four, concerning the surface $T \times T$ embedded in $\mathbb{CP}^8$, where $T$ is a the one dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto $\mathbb{CP}^2$, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.
In this paper we characterise the action of the absolute Galois group on the geometric finite cyclic groups without étale factorization of stack inertia of the profinite geometric fundamental group of moduli spaces of marked curves. As a complementary result, we give the same action on prime order profinite elements in genus 2.
The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of …
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.
The fundamental group of Galois covers of surfaces with octahedral envelope
本文计算了具有4次奇点、退化包络同构于八面体的8次曲面的伽罗瓦覆盖的基本群，确定其为2³³阶亚交换群，通过融合局部群的迭代方法完成计算，并通过三个简化曲面说明非平凡循环对计算的影响。
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.
We study the action of the Galois group on the pro-l-completion of the fundamental group of P^1 - {0, infinity and N-th roots of unity}. We describe the Lie algebra of the image of the Galois action and relate with the geometry of the modular varieties for GL_m for m = 1,2,3,... This story is the l-adic side of the motivic theory of multiple polylogarithms at roots of unity, which generalize the classical cyclotomy theory.
Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X \to S$ this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_η$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T \to X_η$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question to find a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We obtain partial answers to this question, showing for instance that, when $X$ is integral and regu…
We study polystable Higgs bundles twisted by a line bundle over a compact Kähler manifold. These form a Tannakian category when the first and second Chern classes of the bundle are zero. In this paper we identify the corresponding Tannaka group in the case in which the line bundle is of finite order. This group is described in terms of the pro-reductive completion of the fundamental group of the manifold, and the character associated to the line bundle.
Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated to the vector bundles $E$ on $X$ such that $f^*E$ is trivial is equivalent to the category of representations of a finite and etale group scheme. We give a counterexample to this conclusion in the absence of separability.
The Lefschetz Type Theorem For Fundamental Group Schemes
针对向量丛的Tannakian范畴，建立了自然同态忠实平坦、闭浸入或同构的判别准则，并在Langer型正性条件下证明完美域上多种基本群间的同构。
Higgs bundles and fundamental group schemes
本文利用数值有效性概念证明数值平坦希格斯向量丛构成Tannaka范畴，引入希格斯基本群概形，并将其性质与陈类消失猜想联系起来。
Galois Action on Homology of the Heisenberg Curve
本文研究海森堡曲线的基本群与同调结构，构造了阿廷辫子群在其上的作用，并用特征0域上海森堡群的不可约表示描述其同调。
An important piece of information in the theory of the arithmetic Galois action on the geometric fundamental groups of schemes is that divisorial inertia is acted on cyclotomically. We detail in this note the content of this fact in the case of the profinite braid groups arising from complex reflection groups, naturally viewing them as the geometric fundamental groups of the attending classifying spaces. We also include the case of the full (non colored) braid groups, whose completed classifying spaces are Deligne-Mumford stacks rather than schemes.
Exact sequences of fundamental group schemes for $G$-torsors
该研究解决Tannakian基本群概型的实现问题，结合正合序列、Godeaux--Serre构造及Lefschetz型定理，实现了两类群概型对应的光滑射影簇基本群概型。
For the Galois closure $\Xgal$ of a generic projection from a surface $X$, it is believed that $π_1(\Xgal)$ gives rise to new invariants of $X$. However, in all examples this group is surprisingly simple. In this article, we offer an explanation for this phenomenon: We compute a quotient of $π_1(\Xgal)$ that depends on $π_1(X)$ and data from the generic projection only. In all known examples except one, this quotient is in fact isomorphic to $π_1(\Xgal)$. As a byproduct, we simplify part of the computations of Moishezon, Teicher and others.
Cohomology of the differential fundamental group of algebraic curves
本文研究特征零域上光滑射影曲线的微分基本群，通过Tannakian对偶定义，探讨带联络向量丛的de Rham上同调与对应表示的群上同调的关系，并得到消没与非消没结果。
Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the …
Let $X$ be an algebraic surface of degree $5$, which is considered as a branch cover of $\mathbb{CP}^2$ with respect to a generic projection. The surface has a natural Galois cover with Galois group $S_5$. In this paper, we deal with the fundamental groups of Galois covers of degree $5$ surfaces that degenerate to nice plane arrangements; each of them is a union of five planes such that no three planes meet in a line.
For a class of affine algebraic groups $\mathcal C$ over a field, we define the notions of $\mathcal C$-fundamental gerbe of a fibered category, generalizing what we had done in arXiv:1204.1260 for finite group schemes. We give sufficient conditions on $\mathcal C$ implying that a fibered category $X$ over $κ$ satisfying mild hypotheses admits a Nori $\mathcal C$-fundamental gerbe. We show that these are verified in particular by the classes of virtually abelian and virtually unipotent group schemes. In the second situation, under a properness condition on $X$, we give a tannakian interpretati…
We propose two new approaches to the Tannakian Galois groups of holonomic D-modules on abelian varieties. The first is an interpretation in terms of principal bundles given by the Fourier-Mukai transform, which shows that they are almost connected. The second constructs a microlocalization functor relating characteristic cycles to Weyl group orbits of weights. This explains the ubiquity of minuscule representations, and we illustrate it with a Torelli theorem and with a bound for decompositions of a given subvariety as a sum of subvarieties. The appendix sketches a twistor variant that may be …
Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties
本研究将p-进局部域绝对伽罗瓦群表示的Sen算子构造推广到带半稳定图的p-进仿射簇基本群，证明其模的典范性，结合p-进辛普森对应描述李代数作用并延拓到无限维表示，还证明几何Sen算子零化局部解析向量。
In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based …
The Künneth Formula Of Fundamental Group Schemes
针对proper态射，给出Tannakian基本群概形同伦序列正合的统一判据，进而得到两个proper连通概形乘积的基本群概形Künneth公式的等价条件，并应用于多种基本群概形。
Let $X$ be a surface of degree $n$, projected onto $\mathbb{CP}^2$. The surface has a natural Galois cover with Galois group $S_n.$ It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of $X.$ In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree $n \leq 4$, that degenerate to a nice plane arrangement, namely a union of $n$ planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree $4$ em…
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