Rational points of quiver moduli spaces

Hoskins, Victoria; Schaffhauser, Florent

doi:10.5802/aif.3334

Hoskins, Victoria ; Schaffhauser, Florent

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

Abstract
Résumé

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 $\bar{k}$ -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 $\bar{k}$ -representations of a quiver are definable over a certain central division algebra over their field of moduli.

Etant donné un corps parfait $k$ et une clôture algébrique $\bar{k}$ de $k$ , les espaces de modules de $\bar{k}$ -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 $\bar{k}$ -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.

Received: 2018-04-10
Revised: 2019-03-07
Accepted: 2019-09-18
Published online: 2020-06-26

DOI: https://doi.org/10.5802/aif.3334
Classification: 14D20, 14L24, 16G20
Keywords: Algebraic moduli problems, Geometric Invariant Theory, Representations of quivers

@article{AIF_2020__70_3_1259_0,
     author = {Hoskins, Victoria and Schaffhauser, Florent},
     title = {Rational points of quiver moduli spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {1259-1305},
     doi = {10.5802/aif.3334},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1259_0/}
}

Hoskins, Victoria; Schaffhauser, Florent. Rational points of quiver moduli spaces. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1259-1305. doi : 10.5802/aif.3334. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1259_0/

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