no. 3

Rational points of quiver moduli spaces  [ Points rationnels des variétés de carquois ]
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1259-1305.

Etant donné un corps parfait k et une clôture algébrique k ¯ de k, les espaces de modules de 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 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.

Avertissement :

Suite à une erreur de la rédaction, il manque dans la version publiée de cet article des informations sur les financements du second auteur. Le second auteur est soutenu par la Convocatoria 2018-2019 de la Facultad de Ciencias (Uniandes), Programa de investigación « Geometría y Topología de los Espacios de Módulos », le programme de recherche et d’innovation Horizon 2020 de l’Union européenne (subvention n°795222), et L’Institut d’Études Avancées de l’Université de Strasbourg (USIAS).

Compléments :
Des compléments sont fournis pour cet article dans le fichier séparé :

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 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 k ¯-representations of a quiver are definable over a certain central division algebra over their field of moduli.

Disclaimer:

Due to an editorial error, the published version of this article lacks information on the second author’s funding. The second author is supported by Convocatoria 2018-2019 de la Facultad de Ciencias (Uniandes), Programa de investigación “Geometría y Topología de los Espacios de Módulos”, the European Union’s Horizon 2020 research and innovation programme under grant agreement No 795222 and the University of Strasbourg Institute of Advanced Study (USIAS).

Supplementary Materials:
Supplementary material for this article is supplied as a separate file:

Reçu le : 2018-04-10
Révisé le : 2019-03-07
Accepté le : 2019-09-18
Publié le : 2020-06-26
DOI : https://doi.org/10.5802/aif.3334
Classification : 14D20,  14L24,  16G20
Mots clés : Problèmes de modules en géométrie algébrique, Théorie Géométrique des Invariants,Représentations de carquois 
@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},
     pages = {1259--1305},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     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, Tome 70 (2020) no. 3, pp. 1259-1305. doi : 10.5802/aif.3334. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1259_0/

[1] Baily, Walter. L. Jun On the theory of θ-functions, the moduli of abelian varieties, and the moduli of curves, Ann. Math., Volume 75 (1962), pp. 342-381 | Article | MR 0162799 | Zbl 1432.33001

[2] Baily, Walter. L. Jun On the theory of automorphic functions and the problem of moduli, Bull. Am. Math. Soc., Volume 69 (1963), pp. 727-732 | Article | MR 0156002 | Zbl 0920.33001

[3] Le Bruyn, Lieven Representation stacks, D-branes and noncommutative geometry, Commun. Algebra, Volume 40 (2012) no. 10, pp. 3636-3651 | Article | MR 2982885 | Zbl 1260.14005

[4] Căldăraru, Andrei H. Derived categories of twisted sheaves on Calabi–Yau manifolds (2000) (Ph. D. Thesis) | Article | MR 969211 | Zbl 0661.10038

[5] Gille, Philippe; Szamuely, Tamás Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, Volume 101, Cambridge University Press, 2006, xii+343 pages | Article | MR 2266528 | Zbl 0722.11029

[6] Görtz, Ulrich; Wedhorn, Torsten Algebraic geometry I. Schemes. With examples and exercises, Advanced Lectures in Mathematics, Vieweg + Teubner, 2010, viii+615 pages | Article | MR 2675155 | Zbl 1305.32007

[7] Huggins, Bonnie Fields of moduli and fields of definition of curves (2005) (Ph. D. Thesis) | Article | MR 1849551 | Zbl 0983.41020

[8] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010, xviii+325 pages | Article | MR 892317 | Zbl 0618.10029

[9] de Jong, A. J. A result of Gabber, 2004 (http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf) | Article | MR 884797 | Zbl 0602.58044

[10] King, Alastair D. Moduli of Representations of Finite Dimensional Algebras, Q. J. Math., Oxf. II. Ser., Volume 45 (1994), pp. 515-530 | Article | Numdam | MR 2518081 | Zbl 1183.14038

[11] Koizumi, Shoji The fields of moduli for polarized abelian varieties and for curves, Nagoya Math. J., Volume 48 (1972), pp. 37-55 | Article | MR 0352095 | Zbl 1168.30022

[12] Langton, Stacy Guy Valuative criteria for families of vector bundles on algebraic varieties, Ann. Math., Volume 101 (1975), pp. 88-110 | Article | MR 364255 | Zbl 0918.65002

[13] Lieblich, Max Moduli of twisted sheaves, Duke Math. J., Volume 138 (2007) no. 1, pp. 23-118 | Article | MR 2309155 | Zbl 1177.14052

[14] Mumford, David B.; Fogarty, John C.; Kirwan, Frances Clare Geometric Invariant Theory, Springer, 1993 | MR 711197 | Zbl 0543.10020

[15] Olsson, Martin Algebraic spaces and stacks, Colloquium Publications, Volume 62, American Mathematical Society, 2016, xi+298 pages | Article | MR 3495343 | Zbl 1025.58012

[16] Ramanan, Sundararaman Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci., Math. Sci., Volume 90 (1981) no. 2, pp. 151-166 | Article | MR 1284924 | Zbl 0788.30003

[17] Reineke, Markus Moduli of representations of quivers, Trends in representation theory of algebras and related topics (EMS Series of Congress Reports), European Mathematical Society, 2008, pp. 589-637 | Article | MR 1370116 | Zbl 0973.58016

[18] Reineke, Markus; Schröer, Stefan Brauer groups for quiver moduli, Algebr. Geom., Volume 4 (2017) no. 4, pp. 452-471 | Article | MR 1394748 | Zbl 0872.58062

[19] Romagny, Mathieu; Wewers, Stefan Hurwitz spaces, Groupes de Galois arithmétiques et différentiels (Séminaires et Congrès) Volume 13, Société Mathématique de France, 2006, pp. 313-341 | Article | MR 2316356 | Zbl 1091.11034

[20] Schaffhauser, Florent Real points of coarse moduli schemes of vector bundles on a real algebraic curve, J. Symplectic Geom., Volume 10 (2012) no. 4, pp. 503-534 | Article | MR 2069336 | Zbl 1054.11044

[21] Sekiguchi, Tsu Wild ramification of moduli spaces for curves or for abelian varieties, Compos. Math., Volume 54 (1985) no. 3, pp. 331-372 | Article | Numdam | MR 791506 | Zbl 1347.11065

[22] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, Volume 67, Springer, 1979, viii+241 pages (Translated from the French by Marvin Jay Greenberg) | Zbl 0362.10022

[23] Seshadri, Conjeeveram Srirangachari Space of unitary vector bundles on a compact Riemann surface, Ann. Math., Volume 85 (1967), pp. 303-336 | MR 233371 | Zbl 1087.01018

[24] Seshadri, Conjeeveram Srirangachari Geometric reductivity over arbitrary base, Adv. Math., Volume 26 (1977) no. 3, pp. 225-274 | Article | MR 0466154 | Zbl 0041.42701

[25] Shimura, Goro On analytic families of polarized abelian varieties and automorphic functions, Ann. Math., Volume 78 (1963), pp. 149-192 | Article | MR 0156001 | Zbl 1111.58026

[26] Stacks Project Authors Stacks Project, 2017 (http://stacks.math.columbia.edu) | Article | MR 1172692 | Zbl 0772.58063

[27] Tamme, Günter Introduction to étale cohomology, Universitext, Springer, 1994, x+186 pages (Translated from the German by Manfred Kolster) | Article | MR 1317816 | Zbl 0947.58025