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 M be a 1-motive defined over a field of characteristic 0. To M we can associate its motivic Galois group, G_mot(M), which is the geometrical interpretation of the Munford-Tate group of M. We prove that the unipotent radical of the Lie algebra of G_mot(M) is the semi-abelian variety defined by the adjoint action of the semi-simplified of the Lie algebra of G_mot(M) on itself.
We show that the statement analogous to the Mumford-Tate conjecture for abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group. Contains an appendix by P. Deligne. In this version: Minor corrections here and there. The Main result about the image of the Galois representations is now enriched by an adelic statement.
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…
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…
Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without non…
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…
In this thesis, two $\bar{\mathbb{Q}}_\ell$-local systems, $\vphantom{\mathcal{E}}^\circ \mathcal{E}$ and $\vphantom{E}^\circ \mathcal{E}^\prime$ on the regular unipotent subvariety $\mathcal{U}_{0,K}$ of $p$-adic $\operatorname{SL}_2(K)$ are constructed. Making use of the equivalence between $\bar{\mathbb{Q}}_\ell$-local systems and $\ell$-adic representations of the étale fundamental group, we prove that these local systems are equivariant with respect to conjugation by $\operatorname{SL}_2(K)$ and that their nearby cycles, when taken with respect to appropriate integral models, descend to l…
Dipartimento di Matematica “Tullio Levi-Civita”（数学系“图利奥·莱维-奇维塔”）
In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(φ,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$-adic analogue of Oda's theorem that a semistable curve over a $p$-adic field has good reduction iff the Galois action on its $\ell$-adic unipotent fundamental group is unramified.
Let $X$ be a variety (possibly non-complete or singular) over a finitely generated field $k$ of characteristic $0$. For a prime number $\ell$, let $ρ_\ell$ be the Galois representation on the first $\ell$-adic cohomology of $X$. We show that if $\ell$ varies the image of $ρ_\ell$ is of bounded index in the group of $\mathbb{Z}_\ell$-points of its Zariski closure. We use this to improve a recent result of Litt about arithmetic representations of geometric fundamental groups. Litt's result says that there exist constants $N = N(X,\ell)$ such that every arithmetic representation $π_1(X_{\bar{k}})…
We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider sim…
Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtr…
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…
We propose a generalization of Ledet conjecture, which predicts the essential dimension of cyclic $p$-groups in characteristic $p$, for finite commutative unipotent group schemes. And we show some evidence and some consequences of this new conjecture.
In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta-Iovita-Kim's article: obtaining in this way a complete algebraic criterion for good reduction for curves.
It is a theorem of Kim-Tamagawa that the $\mathbb Q_\ell$-pro-unipotent Kummer map associated to a smooth projective curve $Y$ over a finite extension of $\mathbb Q_p$ is locally constant when $\ell\neq p$. The present paper establishes two generalisations of this result. Firstly, we extend the Kim-Tamagawa Theorem to the case that $Y$ is a smooth variety of any dimension. Secondly, we formulate and prove the analogue of the Kim-Tamagawa Theorem in the case $\ell = p$, again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale-de Rham comparison theorem for …
We study the unipotent completion $Π^{DR}_{un}(x_0, x_1, X_K)$ of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space $Π^{DR}_{un}(x_0, x_1, X_K)$ possesses a distinguished element. In the other words, given a vector bundle E on $X_K$ together with a unipotent integrable connection, we have {\sf a canonical} isomorphism $E_{x_0}\simeq E_{x_1}$ between the fibers. The latter construction is a generalization of Colmez's p-adic integration (rk E=2) and Coleman's p-adic iterated integrals ($X_K$…
This is a long overdue write up of the following: If the fundamental group of a normal complex algebraic variety (respectively Zariski open subset of a compact Kähler manifold) is a solvable group of matrices over Q (respectively polycyclic group), then it is virtually nilpotent. This should be interpreted as saying that a large class of solvable groups will not occur as fundamental groups of such spaces. The essential strategy is to first "complete" these groups to algebraic groups (generalizing constructions of Malcev and Mostow), and then check that the identity component is unipotent. This…
By developing a theory of deformations over nilpotent Lie algebras, based on Schlessinger's deformation theory over Artinian rings, this paper investigates the pro-l-unipotent fundamental group of a variety X. If X is smooth and proper, defined over a finite field, then the Weil conjectures imply that this group is quadratically presented. If X is smooth and non-proper, then the group is defined by equations of bracket length at most four.
We present a generalized version of classical geometric invariant theory à la Mumford where we consider an affine algebraic group $G$ acting on a specific affine algebraic variety $X$. We define the notions of linearly reductive and of unipotent action in terms of the $G$ fixed point functor in the category of $(G,\Bbbk [X])$--modules. In the case that $X=\{\star\}$ we recuperate the concept of lineraly reductive and of unipotent group. We prove in our "relative" context some of the classical results of GIT such as: existence of quotients, finite generation of invariants, Kostant--Rosenlicht's…
The family of all mixed Hodge structures on a given rational vector space $M_\mathbb{Q}$ with a fixed weight filtration $W_\cdot$ and a fixed associated graded Hodge structure $Gr^WM$ is naturally in a one to one correspondence with a complex affine space. We study the unipotent radical of the very general Mumford-Tate group of the family. We do this by using general Tannakian results which relate the unipotent radical of the fundamental group of an object in a filtered Tannakian category to the extension classes of the object coming from the filtration. Our main result shows that if $Gr^WM$ i…
We present an averaging process for sections of a torsor under a unipotent group. This process allows one to integrate local sections of such a torsor into a global simplicial section. The results of this paper have applications to deformation quantization of algebraic varieties.
For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for G^\vee-modules, where G^\vee is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to G…
Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.
