Tame parahoric nonabelian Hodge correspondence in positive characteristic over algebraic curves

• Mao Li ^[2] ; Hao Sun ^[1]
  1. [1] South China University of Technology

     South China University of Technology

     China

     
  2. [2] Department of Mathematics, University of Illinois at Urbana-Champaign, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-36
• Idioma: inglés
• DOI: 10.1007/s00029-024-00954-2
• Enlaces
  • Texto completo
• Resumen

  • Let G be a reductive group, and let X be an algebraic curve over an algebraically closed field k with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame G-local systems over X and logarithmic G-Higgs bundles over the Frobenius twist X . To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence.

• Referencias bibliográficas
  • Abramovich, D., Olsson, M., Vistoli, A.: Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58(4), 1057–1091 (2008).
  • Balaji, V., Biswas, I., Pandey, Y.: Connections on parahoric torsors over curves. Publ. Res. Inst. Math. Sci. 53(4), 551–585 (2017).
  • Balaji, V., Seshadri, C.S.: Moduli of parahoric G-torsors on a compact Riemann surface. J. Algebr. Geom. 24, 1–49 (2015).
  • Biquard, O., García-Prada, O., Mundet i Riera, I.: Parabolic Higgs bundles and representations of the fundamental group of a punctured surface...
  • Boalch, P.: Riemann–Hilbert for tame complex parahoric connections. Transform. Groups 16, 27–50 (2011).
  • Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes...
  • Chen, T.H., Zhu, X.: Non-abelian Hodge theory for algebraic curves in characteristic p. Geom. Funct. Anal. 25, 1706–1733 (2015).
  • Chen, T.H., Zhu, X.: Geometric Langlands in prime characteristic. Compos. Math. 153(2), 395–452 (2017).
  • Chernousov, V., Gille, P., Pianzola, A.: Torsors over the punctured affine line. Am. J. Math. 134(6), 1541–1583 (2012).
  • Donagi, R., Gaitsgory, D.: The gerbe of Higgs bundles. Transform. Groups 7, 109–153 (2002).
  • Herrero, A.F.: Reduction theory for connections over the formal punctured disc (2020). arXiv:2003.00008.
  • Huang, P., Kydonakis, G., Sun, H., Zhao, L.: Tame parahoric nonabelian Hodge correspondence on curves (2022). arXiv:2205.15475.
  • Huang, P., Sun, H.: Meromorphic parahoric Higgs torsors and filtered Stokes G-local systems on curves. Adv. Math. 429, 109183 (2023).
  • Kydonakis, G., Sun, H., Zhao, L.: Logahoric Higgs torsors for a complex reductive group. Math. Ann. 388(3), 3183–3228 (2024).
  • Li, M.: The Poincaré line bundle and autoduality of Hitchin fibers (2020). arXiv:2005.01396.
  • Ogus, A.: F-crystals, Griffiths transversality, and the Hodge decomposition. Astérisque, no. 221, ii+183 pp (1994).
  • Ogus, A., Vologodsky, V.: Nonabelian Hodge theory in characteristic p. Publ. Math. Inst. Hautes Étud. Sci. 106, 1–138 (2007).
  • Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219(1), 118–198 (2008).
  • Shen, S.: Tamely ramified geometric Langlands correspondence in positive characteristic. arXiv:1810.12491.
  • Simpson, C.T.: Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3(3), 713–770 (1990).
  • Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math. 75, 5–95 (1992).
  • Sun, H.: Moduli problem of Hitchin pairs over Deligne–Mumford stacks. Proc. Am. Math. Soc. 150(1), 131–143 (2022).
  • Sun, H.: Moduli space of D-modules on projective Deligne–Mumford stacks (2020). arXiv:2003.11674.
  • Teleman, C., Woodward, C.: Parabolic bundles, products of conjugacy classes, and quantum cohomology. Ann. L’Inst. Fourier 3, 713–748 (2003).
  • Yun, Z.: Global Springer theory. Adv. Math. 228, 266–328 (2011).