Rational points of quiver moduli spaces

[Points rationnels des variétés de carquois]

Hoskins, Victoria¹ ; Schaffhauser, Florent²

¹ Freie Universität Berlin Arnimallee 3, Raum 011 14195 Berlin, Germany
² Universidad de Los Andes Carrera 1 #18A-12 111 711 Bogotá, Colombia

Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1259-1305.

Abstract (VO)

For a perfect base field $k$, we investigate arithmetic aspects of moduli spaces of quiver representations over $k$: we study actions of the absolute Galois group of $k$ on the $\overline{k}$-valued points of moduli spaces of quiver representations over $k$ and we provide a modular interpretation of the fixed-point set using quiver representations over division algebras, which we reinterpret using moduli spaces of twisted quiver representations (we show that those spaces provide different $k$-forms of the initial moduli space of quiver representations). Finally, we obtain that stable $\overline{k}$-representations of a quiver are definable over a certain central division algebra over their field of moduli.

Résumé

Etant donné un corps parfait $k$ et une clôture algébrique $\overline{k}$ de $k$, les espaces de modules de $\overline{k}$-représentations semistables d’un carquois $Q$ sont des $k$-variétés algébriques dont nous étudions ici les propriétés arithmétiques, en particulier les points rationnels et leur interprétation modulaire. Outre les représentations à coefficients dans $k$, apparaissent naturellement certaines représentations rationnelles dites tordues, à coefficients dans une algèbre à division définie sur $k$ et qui donnent lieu à différentes $k$-formes de la variété des modules initiale. En guise d’application, on montre qu’une $\overline{k}$-représentation stable du carquois $Q$ est définissable sur une algèbre à division centrale bien précise, elle-même définie sur le corps des modules de la représentation considérée.

Détail

Reçu le : 2018-04-10
Révisé le : 2019-03-07
Accepté le : 2019-09-18
Publié le : 2020-06-26

DOI : 10.5802/aif.3334

Classification : 14D20, 14L24, 16G20
Keywords: Algebraic moduli problems, Geometric Invariant Theory, Representations of quivers
Mots-clés : Problèmes de modules en géométrie algébrique, Théorie Géométrique des Invariants, Représentations de carquois

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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