Algebraic complete integrability of an integrable system of Beauville
Annales de l'Institut Fourier, Volume 58 (2008) no. 2, p. 559-570

We show that the Beauville’s integrable system on a ten dimensional moduli space of sheaves on a K3 surface constructed via a moduli space of stable sheaves on cubic threefolds is algebraically completely integrable, using O’Grady’s construction of a symplectic resolution of the moduli space of sheaves on a K3.

Nous montrons que le système intégrable de Beauville sur un espace de dimension dix de modules de faisceaux sur une surface K3 construit par un espace de modules de faisceaux stables sur les cubiques de dimension trois est algébriquement complètement intégrable. Nous utilisons la construction d’O’Grady d’une résolution symplectique de l’espace des modules de faisceaux sur une surface K3.

DOI : https://doi.org/10.5802/aif.2360
Classification:  14J60,  37J35
Keywords: Integrable system, moduli space of stable sheaves
@article{AIF_2008__58_2_559_0,
     author = {Hwang, Jun-Muk and Nagai, Yasunari},
     title = {Algebraic complete integrability of an integrable system of Beauville},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {559-570},
     doi = {10.5802/aif.2360},
     mrnumber = {2410382},
     zbl = {1144.14037},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_559_0}
}
Hwang, Jun-Muk; Nagai, Yasunari. Algebraic complete integrability of an integrable system of Beauville. Annales de l'Institut Fourier, Volume 58 (2008) no. 2, pp. 559-570. doi : 10.5802/aif.2360. http://www.numdam.org/item/AIF_2008__58_2_559_0/

[1] Beauville, Arnaud Vector bundles on the cubic threefold, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 312 (2002), pp. 71-86 | MR 1941574 | Zbl 1056.14059

[2] Druel, Stéphane Espace des modules de faisceaux de rang 2 semi-stables de classes de Chern c 1 =0,c 2 =2 et c 3 =0 sur la cubique de 4 , Internat. Math. Res. Notices, Tome 19 (2000), pp. 985-1004 | Article | MR 1792346 | Zbl 1024.14004

[3] Huybrechts, Daniel; Lehn, Manfred The Geometry of Moduli Spaces of Sheaves, Friedr. Vieweg & Sohn, Braunschweig, Aspects of Mathematics, E31 (1997) | MR 1450870 | Zbl 0872.14002

[4] Kirwan, Frances Clare Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2), Tome 122 (1985) no. 1, pp. 41-85 | Article | MR 799252 | Zbl 0592.14011

[5] Lehn, Manfred; Sorger, Christoph La singularité de O’Grady, J. Alg. Geom., Tome 15 (2006) no. 4, pp. 753-770 | Article | Zbl 05141406

[6] Mukai, Shigeru Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., Tome 77 (1984) no. 1, pp. 101-116 | Article | MR 751133 | Zbl 0565.14002

[7] O’Grady, Kieran G. Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math., Tome 512 (1999), pp. 49-117 | Article | Zbl 0928.14029