no. 3
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.

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 :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3334
Classification : 14D20, 14L24, 16G20
Keywords: Algebraic moduli problems, Geometric Invariant Theory, Representations of quivers
Mot clés : Problèmes de modules en géométrie algébrique, Théorie Géométrique des Invariants, Représentations de carquois
Hoskins, Victoria 1 ; Schaffhauser, Florent 2

1 Freie Universität Berlin Arnimallee 3, Raum 011 14195 Berlin, Germany
2 Universidad de Los Andes Carrera 1 #18A-12 111 711 Bogotá, Colombia
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@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{\textquoteright}institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     doi = {10.5802/aif.3334},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3334/}
}
TY  - JOUR
AU  - Hoskins, Victoria
AU  - Schaffhauser, Florent
TI  - Rational points of quiver moduli spaces
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 1259
EP  - 1305
VL  - 70
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3334/
DO  - 10.5802/aif.3334
LA  - en
ID  - AIF_2020__70_3_1259_0
ER  - 
%0 Journal Article
%A Hoskins, Victoria
%A Schaffhauser, Florent
%T Rational points of quiver moduli spaces
%J Annales de l'Institut Fourier
%D 2020
%P 1259-1305
%V 70
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3334/
%R 10.5802/aif.3334
%G en
%F 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/articles/10.5802/aif.3334/

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[4] Căldăraru, Andrei H. Derived categories of twisted sheaves on Calabi–Yau manifolds, Ph. D. Thesis, Cornell University, USA (2000) | DOI | MR | Zbl

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

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

[7] Huggins, Bonnie Fields of moduli and fields of definition of curves, Ph. D. Thesis, University of California, Berkeley, USA (2005) | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

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

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

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

Cité par Sources :