Tannakian Formalism Research Articles

Perverse 𝔽 p -sheaves on the affine Grassmannian

Abstract For a reductive group over an algebraically closed field of characteristic p > 0 {p>0} we construct the abelian category of perverse 𝔽 p {\mathbb{F}_{p}} -sheaves on the affine Grassmannian that are equivariant with respect to the action of the positive loop group. We show this is a symmetric monoidal category, and then we apply a Tannakian formalism to show this category is equivalent to the category of representations of a certain affine monoid scheme. We also show that our work provides a geometrization of the inverse of the mod p Satake isomorphism. Along the way we prove that affine Schubert varieties are globally F-regular and we apply Frobenius splitting techniques to the theory of perverse 𝔽 p {\mathbb{F}_{p}} -sheaves.

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

It has been proven by Serre, Larsen–Pink and Chin, that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have “the same” pi _0 and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent F-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent F-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent F-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects.

A motivic Grothendieck–Teichmüller group

This paper proves the Beilinson-Soul{\'e} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches groups to these categories and morphisms reflecting the functorialities leading to the definition of a motivic Grothendieck-Teichm{\"u}ller group. Proofs of the above properties rely on the geometry of the tower of the moduli spaces . This allows us to treat the general case of motives over Spec(Z) with integral coefficients working in Spitzweck's category of motives. From there, passing to Q-coefficients we deal with the classical tannakian formalism and explain how working over Spec(Q) allows a more concrete description of the tannakian group.

The Galois group of the category of mixed Hodge–Tate structures

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 the Hopf algebra H. Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R over a field k, equipped with an invertible line k(1). When R is the sheaf of algebras given by the holomorphic de Rham complex of a complex manifold X, and the line is Q(1), the related Hopf dg-algebra describes a dg-model of the derived category of variations of Hodge-Tate structures on X. Its cobar complex provides a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, which we relate to the p-adic Hodge theory.

Tannakian formalism for fiber functors over tensor categories

In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category of comodules over a Hopf algebra in the target category. We will also give a description of this Hopf algebra using the notion of framed objects.

Adelic descent theory

A result of André Weil allows one to describe rank$n$vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set$\text{GL}_{n}(\mathbb{A})$of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to$G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles$\mathbb{A}_{X}^{\bullet }$, we have an equivalence$\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$between perfect complexes on$X$and cartesian perfect complexes for$\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal$\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers

In this article we develop a broad generalization of the classical Bost–Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck’s category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost–Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

Modified Mixed Realizations, New Additive Invariants, and Periods of DG Categories

To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, etale, Hodge, etc) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of dg categories. This leads to new additive invariants, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism.

Noncommutative numerical motives, Tannakian structures, and motivic Galois groups

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. 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 \overline{HP_\ast} on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues C_{NC} and D_{NC} of Grothendieck's standard conjectures C and D . Assuming C_{NC} , we prove that NNum (k)_F can be made into a Tannakian category NNum ^\dagger(k)_F by modifying its symmetry isomorphism constraints. By further assuming D_{NC} , we neutralize the Tannakian category Num ^\dagger(k)_F using \overline{HP_\ast} . Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups . Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich.

Cubic threefolds, Fano surfaces and the monodromy of the Gauss map

The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.

Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

This article is the sequel to (Marcolli and Tabuada in Sel Math 20(1):315–358, 2014). We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub’s weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin–Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub’s weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky’s category of mixed motives.

On the Tannaka group attached to the Theta divisor of a generic principally polarized abelian variety

To any closed subvariety $Y$ of a complex abelian variety one can attach a reductive algebraic group $G$ which is determined by the decomposition of the convolution powers of $Y$ via a certain Tannakian formalism. For a theta divisor $Y$ on a principally polarized abelian variety, this group $G$ provides a new invariant that naturally endows the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ with a finite constructible stratification. We determine $G$ for a generic principally polarized abelian variety, and for $g=4$ we show that the stratification detects the locus of Jacobian varieties inside the moduli space of abelian varieties.

Model theory and the Tannakian formalism

We draw the connection between the model theoretic notions of internality and the binding group on one hand, and the Tannakian formalism on the other. More precisely, we deduce the fundamental results of the Tannakian formalism by associating to a Tannakian category a first order theory, and applying the results on internality there. We then formulate the notion of a differential tensor category, which axiomatises the category of differential representations of differential linear groups, and show how the model theoretic techniques can be used to deduce the analogous results in that context.

Perverse Nori motives

In this work we show that the Tannakian formalism developed by M. Nori applies to representations of quivers with values in an Abelian category that is Noetherian, Artinian and has with finite dimensional Hom groups. As an application, we define a relative version, modeled after perverse sheaves, of the Abelian category of motives constructed by M. Nori over a field of characteristic embedded in the field of complex numbers.

MOTIVIC DYSON–SCHWINGER EQUATIONS

We consider Dyson–Schwinger Equations (DSEs) in the context of Connes–Kreimer renormalization Hopf algebra of Feynman diagrams and Connes–Marcolli universal Tannakian formalism. This study leads us to formulate a family of Picard–Fuchs equations and a category of Feynman motivic sheaves with respect to each combina torial DSE.

Noncommutative Artin motives

In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category $${{\mathrm{AM}}}(k)_\mathbb Q $$ of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives, we then show that the image of this full embedding, which we call the category $${{\mathrm{NAM}}}(k)_\mathbb Q $$ of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group $$\mathrm{Gal}(\overline{k}/k)$$ from the Tannakian formalism applied to $${{\mathrm{NAM}}}(k)_\mathbb Q $$ . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change, we then construct a short exact sequence relating $$\mathrm{Gal}(\overline{k}/k)$$ with the noncommutative motivic Galois groups of k and $$\overline{k}$$ . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne–Milne. In the mixed world, we introduce the triangulated category $${{\mathrm{NMAM}}}(k)_\mathbb Q $$ of noncommutative mixed Artin motives and construct a faithful functor from the classical category $${{\mathrm{MAM}}}(k)_\mathbb Q $$ of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero $${{\mathrm{NMAM}}}(k)_\mathbb Q $$ is much richer than $${{\mathrm{MAM}}}(k)_\mathbb Q $$ since its higher Ext-groups encode all the (rationalized) higher algebraic $$K$$ -theory of finite etale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne–Milne’s short exact sequence.

Tannakian Formalism Over Fields with Operators

We develop a theory of tensor categories over a endowed with abstract operators. Our notion of a field with operators, coming from work of Moosa and Scanlon, includes the familiar cases of differential and difference fields, Hasse-Schmidt derivations, and their combinations. We develop a corresponding Tannakian formalism, describing the category of representations of linear groups defined over such fields. The paper extends the previously know (classical) algebraic and differential algebraic Tannakian formalisms.

Spherical Varieties and Langlands Duality

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as an algebraic model for the loop space of X. In this paper, we associate to X a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of X such as its compactifications and invariant differential operators. When X is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of G via the (real) geometric Satake correspondence.

Hecke operators on quasimaps into horospherical varieties

Let G be a connected reductive complex algebraic group. This paper and its companion citeGNcombo06 are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G -variety X . The space Z may be thought of as an algebraic model for the loop space of X . The theory we develop associates to X a connected reductive complex algebraic subgroup \check H of the dual group \check G . The construction of \check H is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of \check H . In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical G -variety X_{\mathrm{horo}} , the category Q(Z_{\mathrm{horo}}) is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup \check H_{\mathrm{horo}} is a torus. The case of horospherical varieties may be thought of as a simple example, but it also plays a central role in the general theory. To an arbitrary affine spherical G -variety X , one may associate a horospherical variety X_{\mathrm{horo}} . Its associated subgroup \check H_{\mathrm{horo}} turns out to be a maximal torus in the subgroup \check H associated to X .

P -adic multiple zeta values II. Tannakian interpretations

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 étale 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.