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.

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.

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 0147.39702

[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 0192.26804

[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)

[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 1137.12001

[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 1213.14001

[7] Huggins, Bonnie Fields of moduli and fields of definition of curves (2005) (Ph. D. Thesis) | MR 2708514

[8] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010, xviii+325 pages | Article | Zbl 1206.14027

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

[10] King, Alastair D. Moduli of Representations of Finite Dimensional Algebras, Q. J. Math., Oxf. II. Ser., Volume 45 (1994), pp. 515-530 | Article | MR 1315461 | Zbl 0837.16005

[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 0246.14006

[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 0307.14007

[13] Lieblich, Max Moduli of twisted sheaves, Duke Math. J., Volume 138 (2007) no. 1, pp. 23-118 | Article | MR 2309155 | Zbl 1122.14012

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

[15] Olsson, Martin Algebraic spaces and stacks, Colloquium Publications, Volume 62, American Mathematical Society, 2016, xi+298 pages | Article | MR 3495343 | Zbl 1346.14001

[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 653952 | Zbl 0512.14018

[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 2484736 | Zbl 1206.16009

[18] Reineke, Markus; Schröer, Stefan Brauer groups for quiver moduli, Algebr. Geom., Volume 4 (2017) no. 4, pp. 452-471 | Article | MR 3683503 | Zbl 1400.14035

[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 | MR 2316356 | Zbl 1156.14314

[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 2982021 | Zbl 1408.14045

[21] Sekiguchi, Tsu Wild ramification of moduli spaces for curves or for abelian varieties, Compos. Math., Volume 54 (1985) no. 3, pp. 331-372 | Numdam | MR 791506 | Zbl 0581.14029

[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 0423.12016

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

[24] Seshadri, Conjeeveram Srirangachari Geometric reductivity over arbitrary base, Adv. Math., Volume 26 (1977) no. 3, pp. 225-274 | MR 0466154 | Zbl 0371.14009

[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 0142.05402

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

[27] Tamme, Günter Introduction to étale cohomology, Universitext, Springer, 1994, x+186 pages (Translated from the German by Manfred Kolster) | Article | MR 1317816 | Zbl 0815.14012