Local structure of the Teichmüller and the Riemann moduli stacks

• Doan, An Khuong ^[1]
  1. [1] KU Leuven

     KU Leuven

     Arrondissement Leuven, Bélgica

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 3, 2025, págs. 651-666
• Idioma: inglés
• DOI: 10.1007/s13348-024-00450-y
• Texto completo no disponible (Saber más ...)
• Resumen

  • The goal of this note is to introduce an interesting question proposed by D. Rydh on an analytic version of the local structure of Artin stacks saying that near points with linearly reductive stabilizer, Artin stacks are étale-locally quotient stacks. We give some supporting evidence by verifying it on two fundamental classes of classical analytic moduli spaces: the Teichmüller moduli space and the Riemann moduli space of integrable complex structures whose analytic stack versions have been constructed by a recent work of L. Meersseman.

• Referencias bibliográficas
  • Alper, J., Hall, J., Rydh, D.: A Luna étale slice theorem for algebraic stacks. Annal. Math. 91, 675–738 (2020)
  • Alper, J., Kresch, A.: Equivariant versal deformations of semistable curves. Michigan Math. J. 65(2), 227–250 (2016)
  • Alper, J.: On the local quotient structure of Artin stacks. J. Pure Appl. Algebra 214(9), 1576–1591 (2010)
  • Anschütz, J.: Lecture notes on étale cohomology. Available at https://www.math.uni-bonn.de/people/ja/etcoh/ (2023)
  • Artin, M.: Versal deformations and algebraic stacks. Invent. Math. 27, 165–190 (1969)
  • Arbarello, E., Saccà, G.: Singularities of Bridgeland moduli spaces for K3 categories: an update. arxiv:2307.07789 (2023)
  • Behrend, K., Noah, B.: Uniformization of Deligne-Mumford curves. J. Reine Angew. Math. 599, 111–153 (2006)
  • Brion, M.: Anti-affine algebraic groups. J. Algebra 321(3), 934–952 (2009)
  • Brion, M.: Introduction to actions of algebraic groups. https://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf (2009)
  • Buchdahl, N., Schumacher, G.: An analytic application of geometric invariant theory. J. Geom. Phys. 165, 104237 (2021)
  • Chevalley, C.: Une démonstration d’un théorème sur les groupes algébriques. J. Math. Pures Appl. Neuvième Série 39, 307–317 (1960)
  • Doan, A.K.: A counter-example to the equivariance structure on semi-universal deformation. J. Geom. Anal. 31, 3698–3712 (2020)
  • Doan, A.K.: Equivariant Kuranishi family of complex compact manifolds. Manuscripta Math. 167, 793–808 (2021)
  • Doan, A.K.: Group actions on local moduli space of holomorphic vector bundles. Can. Math. Bull. 66(2), 553–567 (2022)
  • Doan, A.K.: A note on the group extension problem to semi-universal deformation. Indag. Math. 35(2), 240–246 (2024)
  • Dervan, R., Naumann, P.: Moduli space of polarised manifolds via canonical Käller metric. Preprint available at arxiv:1810.02576 (2018)
  • Fan, Y.: Construction of the moduli space of Higgs bundles using analytic methods. Math. Res. Lett. 29(4), 718 (2022)
  • Grothendieck, A.: Techniques de construction en géométrie analytique I-X. Séminaire Henri Cartan Tome 13, 1960–1961 (1960)
  • Hochschild, G., Mostow, G.D.: Representations and representative functions of Lie groups III. Ann. Math. 70(1), 85–100 (1959)
  • Inoue, E.: The moduli space of Fano manifolds with Kähler-Ricci solitons. Adv. Math. 357, 1 (2019)
  • Kodaira, K.: Complex manifolds and deformation of complex structures. In: Classics in Mathematics, English Springer, Berlin (2005)
  • Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures I. Ann. Math. 67, 328–402 (1958)
  • Kuranishi, M.: New Proof for the existence of locally complete families of complex structures. In: Proceedings of the conference on complex...
  • Kuranishi, M.: Deformations of compact complex manifolds. Les Presse de l’Université de Montréal (1971). Séminaire de Mathématiques Supérieures,...
  • Luna, D.: Étales slices. Sur les groupes algébriques, Mémoires de la Société Mathématique de France, 33, 81–105 (1973)
  • Matsushima, Y.: Espaces homogènes de Stein des groupes de Lie complexes. Nagoya Math. J. 16, 205–218 (1960)
  • Meersseman, L.: Kuranishi and Teichmüller. arxiv:1709.07237 (2017)
  • Meersseman, L.: The Teichmüller and Riemann moduli stacks. J. l’École Polytech. Math. 6, 879–945 (2019)
  • Miyajima, K.: Kuranishi family of vector bundles and algebraic description of the moduli spaces of Einstein-Hermitian connections. Publ. Res....
  • Meersseman, L., Nicolau, M.: Deformations and moduli of structures on manifolds: general existence theorem and application to the sasakian...
  • Morimoto, A.: Non-compact complex lie groups without non-constant holomorphic functions. In: Aeppli, A., Calabi, E., Röhrl, H. (eds.), proceedings...
  • Noohi, B.: Foundations of topological stacks I. arxiv:math/0503247 (2005)
  • Odaka, Y., Spotti, C., Sun, S.: Compact moduli spaces of Del Pezzo surfaces and Kähler-Einstein metrics. J. Differ. Geom. 102, 127–172 (2016)
  • Snow, D.M.: Reductive group actions on Stein spaces. Math. Ann. 259, 79–97 (1982)
  • Toën, B.: Higher and derived stacks: a global overview. In: Algebraic geometry - seattle - proceedings symposium pure mathematical 80, 435–487...
  • Wehler, J.: Versal deformation of Hopf surfaces. J. für die reine und angewandte Math. 328, 22–32 (1981)
  • Zung, N.T.: Proper groupoids and momentum maps: linearization, affinity, and convexity. Annal. Sci. de l’École Normale Supérieure Série 4...