Groupes de Brauer algébriques modulo les constants d'espaces homogènes et leurs compactifications

TL;DR

The paper investigates whether the algebraic Brauer group modulo constants, ${\operatorname{Br}_1 X}/{\operatorname{Br} K}$, always identifies with $\mathrm{H}^1(K,\operatorname{Pic}\overline{X})$ for smooth geometrically integral varieties $X$, and whether analogous identifications hold for smooth compactifications. It relates the Hochschild--Serre spectral sequence to obstructions arising from the Springer torsor, showing that the differential $d_2^{0,1}$ coincides with the abelianized Springer obstruction $\eta_X^{\mathrm{ab}}$ and that the elementary obstruction $e(X)$ equals $\eta_X^{\mathrm{ab}}$, yielding a general mechanism to obstruct lifts of abelian torsors. Using this framework, the authors construct $K$-modules with nontrivial $\Sha^1_\omega(K,H')$ and, in particular, produce explicit counterexamples (including $K=\mathbb{C}((t))((t_1))((t_2))$ with $n=4$) showing that ${\operatorname{Br}_1 X}/{\operatorname{Br} K}$ need not equal $\mathrm{H}^1(K,\operatorname{Pic}\overline{X})$, and that for smooth compactifications $X^c$ one gets a proper inclusion ${\operatorname{Br}_1 X^c}/{\operatorname{Br} K} \subsetneq \Sha^1_\omega(K,\operatorname{Pic}\overline{X})$. These results rely on Rost–Voevodsky-type isomorphisms and dualities for higher local fields, highlighting intrinsic limitations of Brauer–Manin-type obstructions in fibrations.

Abstract

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 variété lisse, géométriquement intègre sur un corps $K$. Alors le quotient du groupe Brauer "algébrique" 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 même dans le cas où $X$ est un espace homogène d'un groupe algébrique linéaire connexe sur $K$. Un résultat similaire pour les compactifications lisses de $X$ est aussi donné.

Theorems & Definitions (10)

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• Proposition 2
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• Proposition 6
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• Proposition 8
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• proof : Démosntration du théorème \ref{['thmMain']}