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Charles De Clercq

Publications and source records attributed to Charles De Clercq.

18 recordsLinked to original sources
arXiv2026

Effective Bialynicki-Birula-Brosnan motivic decompositions

Let $G$ be an isotropic reductive group and $X$ be a projective $G$-homogeneous variety. Using results from Bialynicki-Birula, Hesselink and Iversen, Brosnan showed that if $G$ is of inner type, the motive of $X$ can be expressed as a direct sum of Tate twists of motives of projective homogeneous varieties for the anisotropic kernel of $G$. We provide a SageMath implementation of this decomposition, based on a depth-first search of the Cayley graph of the Weyl group of $G$, ensuring the complexity scales with the size of the motive rather than the full Weyl group. As applications, we provide new motivic decompositions for some exceptional groups and show how to extend Karpenko's decompositions for classical groups to characteristic $2$.

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arXiv2025

Rost nilpotence for twisted Milnor hypersurfaces

We show that the strong Rost nilpotence holds for motives of generic hyperplane sections of twisted Milnor hypersurfaces. Hence, we provide a new family of examples of smooth projective algebraic varieties which satisfy the strong Rost nilpotence principle. As an application, we compute the $p$-canonical dimension for such varieties.

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arXiv2025

Smooth profinite groups, I: geometrizing Kummer theory

In this series of three papers, we introduce and study cyclotomic pairs and smooth profinite groups. They are a geometric axiomatisation of Kummer theory for fields, with coefficients $p$-primary roots of unity, for a prime $p$. These coefficients are enhanced, to $G$-linearized line bundles in Witt vectors, over $G$-schemes of characteristic $p$. In the second paper, this upgrade is pushed even further, to the scheme-theoretic setting. In this first article, we introduce cyclotomic pairs, smooth profinite groups and $(G,S)$-cohomology. We prove a first lifting theorem for $G$-linearized torsors under line bundles (Theorem A). With the help of the algebro-geometric tools developed in the second article, this formalism is applied in the third one, to prove the Smoothness Theorem, whose essence reads as follows. Let $G$ be profinite group. Assume that, for every open subgroup $H \subset G$, and for $n=1$, the natural arrow $H^n(H,\mathbb{Z}/p^2) \to H^n(H,\mathbb{Z}/p)$ is surjective. Then, it is also surjective for every such $H$, and every $n \geq 2$. Applied to absolute Galois groups, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology.

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arXiv2025

Smooth profinite groups, III: the Smoothness Theorem

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod $p^2$ Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.

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arXiv2025

A-upper motives of reductive groups

Given a prime number $p$, we perform the study of Chow motives and motivic decompositions, with coefficients in $\mathbb{Z}/p\mathbb{Z}$, of projective homogeneous varieties for $p'$-inner $p$-consistent reductive algebraic groups. Assorted with the known case of $p$-inner reductive groups, our results cover all absolutely simple groups of type not $^3\!D_4$ or $^6\!D_4$, among other examples. First, we define the A-upper motives of such a reductive group $G$; they are indecomposable motives, naturally related to Artin motives built out of spectra of subextensions of a minimal extension over which $G$ become of inner type. With this in hand, we carry on the qualitative study of motivic decompositions for projective $G$-homogeneous varieties. Providing geometric isomorphism criteria for A-upper motives, we obtain a classification of motives of projective $G$-homogeneous varieties, by means of their higher Artin-Tate traces. We also show that the higher Tits $p$-indexes of the group $G$ determine its motivic equivalence class.

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arXiv2024

Higher Tate traces of Chow motives

We establish the complete classification of Chow motives of projective homogeneous varieties for $p$-inner semi-simple algebraic groups, with coefficients in $\mathbb{Z}/p\mathbb{Z}$. Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov-Gille-Merkurjev decompositions and their interpretation through Bialynicki-Birula-Hesselink-Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits $p$-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

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arXiv2023

Lifting vector bundles to Witt vector bundles

Let $X$ be a scheme. Let $r \geq 2$ be an integer. Denote by $W_r(X)$ the scheme of Witt vectors of length $r$, built out of $X$. We are concerned with the question of extending (=lifting) vector bundles on $X$, to vector bundles on $W_r(X)$-promoting a systematic use of Witt modules and Witt vector bundles. To begin with, we investigate two elementary but significant cases, in which the answer to this question is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We then offer a simple (re)formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p>0$, and extend them to reduced $k$-schemes. Some of these results were recently recovered, in another form, by Stefan Schröer. As an application, we prove that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. To conclude, we establish a connection to the work of Zdanowicz, on non-liftability of some projective bundles.

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arXiv2021

Lifting low-dimensional local systems

Let $k$ be a field of characteristic $p>0$. Denote by $W_r(k)$ the ring of truntacted Witt vectors of length $r \geq 2$, built out of $k$. In this text, we consider the following question, depending on a given profinite group $G$. $Q(G)$: Does every (continuous) representation $G\longrightarrow GL_d(k)$ lift to a representation $G\longrightarrow GL_d(W_r(k))$? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in [DCF] under the name "smooth profinite groups". Using Grothendieck-Hilbert' theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over $\mathbb{Z}[\frac{1}{p}]$, smooth curves over algebraically closed fields, and affine schemes over $\mathbb{F}_p$. In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to $Q(G)$, for a cyclotomic profinite group $G$: the answer is positive, when $d=2$ and $r=2$. When $d=2$ and $r=\infty$, we show that any $2$-dimensional representation of $G$ stably lifts to a representation over $W(k)$: see Theorem 6.1. \\When $p=2$ and $k=\mathbb{F}_2$, we prove the same results, up to dimension $d=4$. We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3).

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arXiv2018

Critical varieties and motivic equivalence for algebras with involution

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F_2, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.

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arXiv2015

On the Tits p-indexes of semisimple algebraic groups

The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria of motivic equivalence for semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants and Chow motives of the twisted flag varieties for those groups.

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arXiv2015

Equivalence motivique des groupes algebriques semisimples

Two semisimple algebraic groups are said to be motivic equivalent if the motives of the associated twisted flag varieties are isomorphic modulo any prime p. The purpose of this note is to construct the combinatorial invariants which characterize motivic equivalence and which are the motivic analogues of the Tits indices which appear in the classification of semisimple algebraic groups. The expression of these invariants -the Tits p-indexes- in terms of the classical invariants associated to the natural underlying structures of semisimple algebraic groups allow to produce algebraic criteria of motivic equivalence, generalizing Vishik's criterion of motivic equivalence for the motives of quadrics. It also clarifies the relation between the motives and the rational geometry of twisted flag varieties. Deux groupes semisimples sont dits motiviquement equivalents si les motifs des varietes de drapeaux generalisees associees sont isomorphes modulo tout nombre premier p. L'objet de cette note est de construire les invariants combinatoires qui caracterisent l'equivalence motivique et sont les analogues motiviques des indices de Tits apparaissant dans la classification des groupes algebriques semisimples. L'expression de ces invariants -les p-indices de Tits superieurs- en fonction des indices classiques associes aux structures naturelles sous-jacentes aux groupes semisimples permet de produire des criteres algebriques d'equivalence motivique, generalisant le critere de Vishik d'equivalence motivique des quadriques. Elle permet en outre de clarifier le lien qu'entretiennent les motifs et la geometrie birationnelle des varietes de drapeaux.

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arXiv2013

Motivic equivalence of algebraic groups

Two semisimple algebraic groups of the same type are said to be motivic equivalent if the motives of the associated projective homogeneous varieties of the same type are isomorphic. We give general criteria of motivic equivalence in terms of the so-called higher Tits p-indices of algebraic groups. These results allow to give a complete classification of absolutely simple classical groups up to motivic equivalence in terms of the underlying algebraic structures. Among other applications of this classification, we deduce that there is a bijection between the stable birational equivalence classes of Severi-Brauer varieties of fixed dimension and the motivic equivalence classes of projective linear groups of the same rank.

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arXiv2012

Motivic rigidity of Severi-Brauer varieties

Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi-Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E/F, the motivic decompositions of several Severi-Brauer varieties of D remain the same when extending the scalars to E.

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arXiv2012

A going down theorem for Grothendieck Chow motives

Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of X_E with coefficients in K to be lifted to the base field. This going down result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and Springer.

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arXiv2011

Upper motives of products of projective linear groups

Fix a base field F, a finite field K and consider a sequence of central simple F-algebras A_1,...,A_n. In this note we provide some results toward a classification of the indecomposable motives lying in the motivic decompositions of projective homogeneous varieties under the action of PGL(A_1)x...xPGL(A_n) with coefficients in K. We give a complete classification of those motives if n=1 and derive from it the motivic dichotomy of projective linear groups. We then provide several classification results as well as counterexamples for arbitrary n showing that the situation is less rigid. These results involve a neat study of rational maps between generalized Severi-Brauer varieties which is certainly of independent interest.

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arXiv2011

Classification of upper motives of algebraic groups of inner type A_n

Let A, A' be two central simple algebras over a field F and \mathbb{F} be a finite field of characteristic p. We prove that the upper indecomposable direct summands of the motives of two anisotropic varieties of flags of right ideals X(d_1,...,d_k;A) and X(d'_1,...,d'_s;A') with coefficients in \mathbb{F} are isomorphic if and only if the p-adic valuations of gcd(d_1,...,d_k) and gcd(d'_1,..,d'_s) are equal and the classes of the p-primary components A_p and A'_p of A and A' generate the same group in the Brauer group of F. This result leads to a surprising dichotomy between upper motives of absolutely simple adjoint algebraic groups of inner type A_n

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arXiv2010

Motivic decompositions of projective homogeneous varieties and change of coefficients

We prove that under some assumptions on an algebraic group $G$, indecomposable direct summands of the motive of a projective $G$-homogeneous variety with coefficients in $\mathbb{F}_p$ remain indecomposable if the ring of coefficients is any field of characteristic $p$. In particular for any projective $G$-homogeneous variety $X$, the decomposition of the motive of $X$ in a direct sum of indecomposable motives with coefficients in any finite field of characteristic $p$ corresponds to the decomposition of the motive of $X$ with coefficients in $\mathbb{F}_p$. We also construct a counterexample to this result in the case where $G$ is arbitrary.

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