On automorphisms of semistable G-bundles with decorations
-
Andres Fernandez Herrero
Abstract
We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G-bundles on a smooth projective curve for a reductive algebraic group G. For example, our result applies to the stack of semistable G-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G-bundles. Similar arguments apply to Gieseker semistable G-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decorated G-bundles admitting a quasiprojective good moduli space can be written naturally as a G-linearized global quotient Y/G, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic G-Higgs bundles on a family of curves is smooth over any base in characteristic 0.
-
Communicated by: I. Coskun
Acknowledgements
I would like to thank Mark Andrea de Cataldo, Tomás L. Gómez, Daniel Halpern-Leistner and Nicolas Templier for helpful discussions. I would also like to thank anonymous referees for useful comments on the manuscript.
References
[1] J. Alper: Good moduli spaces for Artin stacks. Ann. Inst. Fourier (Grenoble) 63 (2013), 2349–2402. MR3237451 Zbl 1314.1409510.5802/aif.2833Search in Google Scholar
[2] J. Alper: Adequate moduli spaces and geometrically reductive group schemes. Algebr. Geom. 1 (2014), 489–531. MR3272912 Zbl 1322.1402610.14231/AG-2014-022Search in Google Scholar
[3] J. Alper, D. Halpern-Leistner, J. Heinloth: Existence of moduli spaces for algebraic stacks. Preprint 2018, https://arxiv.org/abs/1812.01128v4Search in Google Scholar
[4] L. Álvarez Cónsul, O. García-Prada: Dimensional reduction, SL(2, C)-equivariant bundles and stable holomorphic chains, Internat. J. Math. 12 (2001), no. 2, 159–201. MR1823573 Zbl 1110.3230510.1142/S0129167X01000745Search in Google Scholar
[5] L. Álvarez Cónsul, O. García-Prada, A. H. W. Schmitt: On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces. IMRP Int. Math. Res. Pap. (2006), Art. ID 73597, 82 pages. MR2253535 Zbl 1111.32012Search in Google Scholar
[6] B. Anchouche, I. Biswas: Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. Amer. J. Math. 123 (2001), 207–228. MR1828221 Zbl 1007.5302610.1353/ajm.2001.0007Search in Google Scholar
[7] I. Biswas, Y. I. Holla: Harder–Narasimhan reduction of a principal bundle. Nagoya Math. J. 174 (2004), 201–223. MR2066109 Zbl 1056.1404610.1017/S0027763000008850Search in Google Scholar
[8] I. Biswas, S. Ramanan: An infinitesimal study of the moduli of Hitchin pairs. J. London Math. Soc. (2) 49 (1994), 219–231. MR1260109 Zbl 0819.5800710.1112/jlms/49.2.219Search in Google Scholar
[9] L. Braun, D. Greb, K. Langlois, J. Moraga: Reductive quotients of klt singularities. Preprint 2021, https://arxiv.org/abs/2111.02812v2Search in Google Scholar
[10] U. Bruzzo, B. Graña Otero: Semistable and numerically effective principal (Higgs) bundles. Adv. Math. 226 (2011), 3655–3676. MR2764901 Zbl 1263.1401910.1016/j.aim.2010.10.026Search in Google Scholar
[11] B. Conrad: Reductive group schemes. In: Autour des schémas en groupes. Vol. I, volume 42/43 of Panor. Synthèses, 93–444, Soc. Math. France, Paris 2014. MR3362641 Zbl 1349.14151Search in Google Scholar
[12] M. Demazure, A. Grothendieck: Schémas en groupes (SGA 3) Tome I. Propriétés générales des schémas en groupes. Lecture Notes in Mathematics, Vol. 151, Springer, Berlin 1970. Revised and annotated edition of the 1970 French original, edited by P. Gille and P. Polo. Documents Mathématiques (Paris), 7. Soc. Math. de France, Paris, 2011. MR2867621 Zbl 1241.14002Search in Google Scholar
[13] T. L. Gómez, A. F. Herrero, A. Zamora: The moduli stack of principal -sheaves. Preprint 2021, https://arxiv.org/abs/2107.03918v4Search in Google Scholar
[14] T. L. Gómez, A. Langer, A. H. W. Schmitt, I. Sols: Moduli spaces for principal bundles in arbitrary characteristic. Adv. Math. 219 (2008), 1177–1245. MR2450609 Zbl 1163.1400810.1016/j.aim.2008.05.015Search in Google Scholar
[15] T. L. Gómez, A. Langer, A. H. W. Schmitt, I. Sols: Moduli spaces for principal bundles in large characteristic. In: Teichmüller theory and moduli problem, volume 10 of Ramanujan Math. Soc. Lect. Notes Ser., 281–371, Ramanujan Math. Soc., Mysore 2010. MR2667560 Zbl 1201.14008Search in Google Scholar
[16] J. Hall, D. Rydh: Coherent Tannaka duality and algebraicity of Hom-stacks. Algebra Number Theory 13 (2019), 1633–1675. MR4009673 Zbl 1423.1401010.2140/ant.2019.13.1633Search in Google Scholar
[17] D. Halpern-Leistner: On the structure of instability in moduli theory. Preprint 2014, https://arxiv.org/abs/1411.0627v5Search in Google Scholar
[18] D. Halpern-Leistner, A. F. Herrero, T. Jones: Moduli spaces of sheaves via affine Grassmannians. Preprint 2021, https://arxiv.org/abs/2107.02172v1Search in Google Scholar
[19] J. Heinloth: Hilbert–Mumford stability on algebraic stacks and applications to G-bundles on curves. Épijournal Geom. Algébrique 1 (2017), Art. 11, 37 pages. MR3758902 Zbl 1410.1401110.46298/epiga.2018.volume1.2062Search in Google Scholar
[20] J. Heinloth, A. H. W. Schmitt: The cohomology rings of moduli stacks of principal bundles over curves. Doc. Math. 15 (2010), 423–488. MR2657374 Zbl 1206.1402910.4171/dm/302Search in Google Scholar
[21] D. Hyeon: Principal bundles over a projective scheme. Trans. Amer. Math. Soc. 354 (2002), 1899–1908. MR1881022 Zbl 1014.1400210.1090/S0002-9947-01-02933-6Search in Google Scholar
[22] J. Le Potier: Systèmes cohérents et structures de niveau. Astérisque no. 214 (1993), 143 pages. MR1244404 Zbl 0881.14008Search in Google Scholar
[23] D. Mumford, J. Fogarty, F. Kirwan: Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2). Springer-Verlag, Berlin 1994. MR1304906 Zbl 0881.1400810.1007/978-3-642-57916-5Search in Google Scholar
[24] S. Ramanan, A. Ramanathan: Some remarks on the instability flag. Tohoku Math. J. (2) 36 (1984), 269–291. MR742599 Zbl 0567.1402710.2748/tmj/1178228852Search in Google Scholar
[25] A. Ramanathan: Stable principal bundles on a compact Riemann surface. Math. Ann. 213 (1975), 129–152. MR369747 Zbl 0284.3201910.1007/BF01343949Search in Google Scholar
[26] A. Ramanathan: Moduli for principal bundles over algebraic curves. I. Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 301–328. MR1420170 Zbl 0284.3201910.1007/BF02867438Search in Google Scholar
[27] A. Ramanathan: Moduli for principal bundles over algebraic curves. II. Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 421–449. MR1425616 Zbl 0901.1400810.1007/BF02837697Search in Google Scholar
[28] A. Schmitt: Moduli for decorated tuples of sheaves and representation spaces for quivers. Proc. Indian Acad. Sci. Math. Sci. 115 (2005), 15–49. MR2120597 Zbl 1076.1401910.1007/BF02829837Search in Google Scholar
[29] A. H. W. Schmitt: Singular principal bundles over higher-dimensional manifolds and their moduli spaces. Int. Math. Res. Not. no. 23 (2002), 1183–1209. MR1903952 Zbl 1034.14017Search in Google Scholar
[30] A. H. W. Schmitt: Moduli problems of sheaves associated with oriented trees. Algebr. Represent. Theory 6 (2003), 1–32. MR1960511 Zbl 1033.14009Search in Google Scholar
[31] A. H. W. Schmitt: Geometric invariant theory and decorated principal bundles. European Mathematical Society, Zürich 2008. MR2437660 Zbl 1159.1400110.4171/065Search in Google Scholar
[32] A. H. W. Schmitt: Notes on coherent systems. Rev. Mat. Teor. Apl. 28 (2021), 1–38. MR4195590 Zbl 7603762Search in Google Scholar
[33] C. T. Simpson: Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. no. 80 (1994), 5–79 (1995). MR1320603 Zbl 0891.1400610.1007/BF02698895Search in Google Scholar
[34] The Stacks Project Authors: Stacks Project, 2022. https://stacks.math.columbia.eduSearch in Google Scholar
[35] R. Steinberg: Regular elements of semisimple algebraic groups. Inst. Hautes Études Sci. Publ. Math. no. 25 (1965), 49–80. MR180554 Zbl 0136.3000210.1007/BF02684397Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Topology of tropical moduli spaces of weighted stable curves in higher genus
- Explicit p-harmonic functions on the real Grassmannians
- Cohomogeneity one central Kähler metrics in dimension four
- Tropical Poincaré duality spaces
- Isometries of wall-connected twin buildings
- On automorphisms of semistable G-bundles with decorations
- Abelian branched covers of rational surfaces
- Polyhedral compactifications, I
Articles in the same Issue
- Frontmatter
- Topology of tropical moduli spaces of weighted stable curves in higher genus
- Explicit p-harmonic functions on the real Grassmannians
- Cohomogeneity one central Kähler metrics in dimension four
- Tropical Poincaré duality spaces
- Isometries of wall-connected twin buildings
- On automorphisms of semistable G-bundles with decorations
- Abelian branched covers of rational surfaces
- Polyhedral compactifications, I
