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 $0\longrightarrow W_pM \longrightarrow M \longrightarrow M/W_pM \longrightarrow 0$: it states that $\underline{\mathfrak{u}}(M)$ is the smallest subobject of $W_{-1}\underline{Hom}(M,M)$ such that the sum of the aforementioned extensions, considered as extensions of $\mathbf{1}$ by $W_{-1}\underline{Hom}(M,M)$, is the pushforward of an extension of $\mathbf{1}$ by $\underline{\mathfrak{u}}(M)$. In this article, we study each of the above-mentioned extensions individually in relation to $\underline{\mathfrak{u}}(M)$. Among other things, we obtain a refinement of Deligne's result, where we give a sufficient condition for when an individual extension $0\longrightarrow W_pM \longrightarrow M \longrightarrow M/W_pM \longrightarrow 0$ is the pushforward of an extension of $\mathbf{1}$ by $\underline{\mathfrak{u}}(M)$. In the second half of the paper, we give an application to mixed motives whose unipotent radical of the motivic Galois group is as large as possible (i.e. with $\underline{\mathfrak{u}}(M)= W_{-1}\underline{Hom}(M,M)$). Using Grothedieck's formalism of \textit{extensions panachées} we prove a classification result for such motives. Specializing to the category of mixed Tate motives we obtain a classification result for 3-dimensional mixed Tate motives over $\mathbb{Q}$ with three weights and large unipotent radicals.
We initiate a study of path spaces in the nascent context of "motivic dga's", under development in doctoral work by Gabriella Guzman. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga, and provides us with a factorization of Kim's relative unipotent section conjecture into several smaller conjectures with a homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that the path spaces of the punctured projective line over a number field are concentrated in degree zero with respect to Levine's t-structure f…
S. Bloch and M. Vlasenko recently introduced a theory of \emph{motivic Gamma functions}, given by periods of the Mellin transform of a geometric variation of Hodge structure, which they tie to the monodromy and asymptotic behavior of certain unipotent extensions of the variation. Here we further examine these Gamma functions and the related \emph{Apéry and Frobenius invariants} of a VHS, and establish a relationship to motivic cohomology and solutions to inhomogeneous Picard-Fuchs equations.
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 study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent étale realisations when the base field is $p$-adic, and in terms of the $\mathbb R$-pro-unipotent Betti--de Rham realisation when the base field is archimedean. In the cours…
In this paper, we establish a link between the structure theory of the pro-unipotent motivic fundamental group of the projective line minus three points and Diophantine geometry. In particular, we give a p-adic proof of Siegel's theorem.
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 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.
The aim of this paper is to study the pro-algebraic fundamental group of a compact Kaehler manifold. Following work by Simpson, the structure of this group's pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.'s result on the de Rham fundamental group, and Goldman and Millson's result on deforming representations of Kaehler groups, and can be regarded as a consequence of formality of the schematic homotopy type. New examples are given…
We define the category of mixed Tate motives over the ring of S-integers of a number field. We define the motivic fundamental group (made unipotent) of a unirational variety over a number field. We apply this to the study of the motivic fundamental group of the projective line punctured at zero, infinity and all N-th roots of unity.
This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret t…
We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of quasi-coherent complexes over a derived quotient stack constructed from a motivic algebra of the abelian variety. We then study the structure of the motivic Galois groups of their mixed motives. We prove that the motivic Galois group is decomposed into a unipotent part constructed from the motivic algebra, and the reductive quotient which is the Tannaka dual of Grothend…
The $G$-representation variety $R_G(Σ_g)$ parametrizes the representations of the fundamental groups of surfaces $π_1(Σ_g)$ into an algebraic group $G$. Taking $G$ to be the groups of $n \times n$ upper triangular or unipotent matrices, we compare two methods for computing algebraic invariants of $R_G(Σ_G)$. Using the geometric method initiated by González-Prieto, Logares and Muñoz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of $R_G(Σ_g)$ in the Grothendieck ring of varieties for $n = 1, \ldots, 5$. Introducing the notion of algebraic representatives we …
In this thesis we develop a Chabauty-Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and therefore Chabauty) in the unipotent case by adding a reductive quotient of the fundamental group. From this perspective, the addition of a reductive part allows one to apply Chabauty-type methods to fundamental groups with trivial unipotent completion, such as $SL_2(\mathbb{Z})$. On the other hand, the unipote…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be…
Let $G$ be a connected reductive algebraic group defined over the finite field $\FF_q$, where $q$ is a power of a good prime for $G$. We write $F$ for the Frobenius morphism of $G$ corresponding to the $\FF_q$-structure, so that $G^F$ is a finite group of Lie type. Let $P$ be an $F$-stable parabolic subgroup of $G$ and $U$ the unipotent radical of $P$. In this paper, we prove that the number of $U^F$-conjugacy classes in $G^F$ is given by a polynomial in $q$, under the assumption that the centre of $G$ is connected. This answers a question of J. Alperin in \cite{alperin}. In order to prove the…
As a higher genus version of universal mixed elliptic motives by Hain and Matsumoto, we consider mixed Teichmüller motives as certain motivic local systems on the moduli space of pointed curves. We show that the category of mixed Teichmüller motives is equivalent to a full subcategory of a certain product category of mixed Tate motives over Z and universal mixed elliptic motives. Furthermore, we show that unipotent fundamental torsors for the universal open curve give rise to a pro-object in the category of mixed Teichmüller motives. Our results can give a realization of motivic correlators pr…
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…
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 $G$ be a simple algebraic group over an algebraically closed field $K$ and let $N = N_G(T)$ be the normalizer of a fixed maximal torus $T\leq G$. Further, let $U$ be the unipotent radical of a fixed Borel subgroup $B$ that contains $T$ and let $U^-$ be the unipotent radical of the opposite Borel subgroup $B^-$. The Bruhat decomposition implies the decomposition $G = NU^-UN$. The Zariski closed subset $U^-U\subset G$ is isomorphic to the affine space $A_K^m$ where $m = \dim G -\dim T$ is the number of roots in the corresponding root system. Here we construct a subgroup $\mathcal{N}\leq \ope…
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration and Ayoub's equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also…
Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi factor of $P$, then $L$ is a Hermitian symmetric subgroup of $G$ and $B$ acts with finitely many orbits both on $\mathfrak{p}^u$ and on $G/L$. In this paper we study the Bruhat order of the $B$-orbits in $\mathfrak{p}^u$ and in $G/L$, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.
In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the archemidean height of the cycle C - C- in the Jacobian of the curve C. We also develop some of the basic theory of rela…
