arXiv subjects

Nguyen Manh Linh

Publications and source records attributed to Nguyen Manh Linh.

7 recordsLinked to original sources

arXiv2026

Insufficiency of the algebraic Brauer--Manin obstruction for homogeneous spaces

Over any number field containing a root of unity of odd prime order, we construct a homogeneous space of $\mathrm{SL}_n$ with finite $2$-nilpotent geometric stabilizers, with a constant unramified algebraic Brauer group, which has no rational point but has local points in every completion of the ground field. This yields the first example of transcendental Brauer--Manin obstruction for homogeneous spaces of connected linear algebraic groups. Our method exploits a previous idea by Borovoi and Kunyavskii.

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arXiv2025

Mayer--Vietoris sequences for complexes of tori

In the patching setting, given a factorization inverse system of fields over which patching for finite-dimensional vector spaces holds, together with a crossed module over the inverse limit field, the corresponding six-term Mayer--Vietoris sequence is constructed, generalizing the classical result of Harbater--Hartmann--Krashen for linear algebraic groups. When the crossed module is a two-term complex of tori, the above sequence is extended into a nine-term exact sequence, notably without any assumption on global domination of Galois cohomology of the inverse system. As an application, we show that patching holds for nonabelian second Galois cohomology of reductive groups with smooth centers. We then obtain a weak local--global principle for this cohomology set in the simply connected semisimple case. We also rediscover a well-known local--global principle for indices of central simple algebras.

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arXiv2023

Non-abelian descent types

We present the notion of non-abelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and non-abelian Galois 2-cohomology due to Springer and Borovoi. The necessity of descent types arises in the context of the descent theory where no torsors are given a priori, for example, when we wish to study the arithmetic properties such as the Brauer--Manin obstruction to the Hasse principle on homogeneous spaces without rational points. This new definition also unifies the types by Colliot-Thélène--Sansuc, the extended types by Harari--Skorobogatov, and the finite descent type by Harpaz--Wittenberg.

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arXiv2023

Sur les espaces homogènes de Borovoi-Kunyavski\uı

We establish the Hasse principle and the weak approximation property for certain homogeneous spaces of $\mathrm{SL}_n$ whose geometric stabilizer is of nilpotency class 2, which were constructed by Borovoi and Kunyavski\uı. These homogeneous spaces verify thus a conjecture of Colliot-Thélène concerning Brauer-Manin obstruction for geometrically rationally connected varieties. -- Nous établissons le principe de Hasse et l'approximation faible pour certains espaces homogènes de $\mathrm{SL}_n$ à stabilisateur géométrique nilpotent de classe 2, construits par Borovoi et Kunyavski\uı. Ces espaces homogènes vérifient donc une conjecture de Colliot-Thélène concernant l'obstruction de Brauer-Manin pour les variétés géométriquement rationnellement connexes.

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arXiv2023

Arithmetics of homogeneous spaces over $p$-adic function fields

Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak approximation problems for homogeneous spaces of $\textrm{SL}_{n,K}$ with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot-Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over $K$, rediscovering the positive answer to the abelian case of the inverse Galois problem over $\mathbb{Q}_p(t)$. In the case where the curve is defined over a higher-dimensional local field instead of a finite extension of $\mathbb{Q}_p$, coarser results are also given.

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arXiv2023

On the descent conjecture for rational points and zero-cycles

The descent method is one of the approaches to study the Brauer--Manin obstruction to the local--global principle and to weak approximation on varieties over number fields, by reducing the problem to ``descent varieties''. In recent lecture notes by Wittenberg, he formulated a ``descent conjecture'' for torsors under linear algebraic groups. The present article gives a proof of this conjecture in the case of connected groups, generalizing the toric case from the previous work of Harpaz--Wittenberg. As an application, we deduce directly from Sansuc's work the theorem of Borovoi for homogeneous spaces of connected linear algebraic groups with connected stabilizers. We are also able to reduce the general case to the case of finite (\'etale) torsors. When the set of rational points is replaced by the Chow group of zero-cycles, an analogue of the above conjecture for arbitrary linear algebraic groups is proved.

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arXiv2022

Groupes de Brauer alg\'ebriques modulo les constants d'espaces homog\`enes et leurs compactifications

Let $X$ be a smooth, geometrically integral variety over a field $K$. Then the quotient of the "algebraic" Brauer group of $X$ by $\operatorname{Br} K$ injects into $\textrm{H}^1(K,\textrm{Pic} \bar{X})$. We show that this inclusion is not always an isomorphism, even in the case where $X$ is a homogeneous space of a connected linear algebraic group over $K$. A similar result for the smooth compactifications of $X$ is also given. ----- Soit $X$ une vari\'et\'e lisse, g\'eom\'etriquement int\`egre sur un corps $K$. Alors le quotient du groupe Brauer "alg\'ebrique" de $X$ par $\operatorname{Br} K$ s'injecte dans $\textrm{H}^1(K,\operatorname{Pic} \bar{X})$. Nous montrons que cette inclusion n'est pas toujours un isomorphisme m\^eme dans le cas o\`u $X$ est un espace homog\`ene d'un groupe alg\'ebrique lin\'eaire connexe sur $K$. Un r\'esultat similaire pour les compactifications lisses de $X$ est aussi donn\'e.

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