Symmetric cubic surfaces and $\mathbf {G}_2$ character varieties
Authors:
Philip Boalch and Robert Paluba
Journal:
J. Algebraic Geom. 25 (2016), 607-631
DOI:
https://doi.org/10.1090/jag/668
Published electronically:
April 6, 2016
MathSciNet review:
3533182
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We will consider a two dimensional “symmetric” subfamily of the four dimensional family of Fricke cubic surfaces. The main result is that such symmetric cubic surfaces arise as character varieties for the exceptional group of type $G_2$. Further, this symmetric family will be related to the fixed points of the triality automorphism of $\operatorname {Spin}(8)$, and an example involving the finite simple group of order $6048$ inside $G_2$ will be considered.
References
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References
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- P. P. Boalch, Riemann-Hilbert for tame complex parahoric connections, Transform. Groups 16 (2011), no. 1, 27–50. MR 2785493 (2012m:14020), DOI https://doi.org/10.1007/s00031-011-9121-1
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- P. P. Boalch, Simply-laced isomonodromy systems, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 1–68. MR 3090254, DOI https://doi.org/10.1007/s10240-012-0044-8
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- N. J. Hitchin, Langlands duality and $G_2$ spectral curves, Q. J. Math. 58 (2007), no. 3, 319–344. MR 2354922 (2008j:14065), DOI https://doi.org/10.1093/qmath/ham016
- K. Iwasaki, A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 131–135. MR 1930217 (2003h:34187)
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- W. Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), no. 1, 91–103. MR 558891 (81a:20043), DOI https://doi.org/10.1007/BF01214715
- Yu. I. Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of $\mathbf {P}^2$, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 186, Amer. Math. Soc., Providence, RI, 1998, pp. 131–151. MR 1732409 (2001b:14086)
- I. Naruki, Cross ratio variety as a moduli space of cubic surfaces, with an appendix by Eduard Looijenga, Proc. London Math. Soc. (3) 45 (1982), no. 1, 1–30. MR 662660 (84d:14020), DOI https://doi.org/10.1112/plms/s3-45.1.1
- A. Oblomkov, Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not. 18 (2004), 877–912. MR 2037756 (2005j:20005), DOI https://doi.org/10.1155/S1073792804133072
- K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\textrm {VI}}$, Ann. Mat. Pura Appl. (4) 146 (1987), 337–381. MR 916698 (88m:58062), DOI https://doi.org/10.1007/BF01762370
- G. W. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), no. 2, 167–191. MR 511189 (80m:14032), DOI https://doi.org/10.1007/BF01403085
- G. W. Schwarz, Invariant theory of $G_{2}$, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 3, 335–338. MR 714998 (85c:20031), DOI https://doi.org/10.1090/S0273-0979-1983-15197-2
- J.-P. Serre, Coordinées de Kac, http://www.college-de-france.fr/site/historique/essai.htm, 2006.
- H. Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Sup. (3) 6 (1889), 3–71 (French). MR 1508833
- D. Yamakawa, Geometry of multiplicative preprojective algebra, Int. Math. Res. Pap. IMRP (2008), Art. ID rpn008, 77 pp. MR 2470573 (2010e:16028)
Additional Information
Philip Boalch
Affiliation:
Laboratoire de Mathématiques d’Orsay (CNRS UMR 8628), Bâtiment 425, Université Paris-Sud, 91405 Orsay, France
MR Author ID:
686227
Email:
philip.boalch@math.u-psud.fr
Robert Paluba
Affiliation:
Laboratoire de Mathématiques d’Orsay (CNRS UMR 8628), Bâtiment 425, Université Paris-Sud, 91405 Orsay, France
Email:
robert.paluba@math.u-psud.fr
Received by editor(s):
June 2, 2013
Received by editor(s) in revised form:
July 9, 2014, and September 27, 2014
Published electronically:
April 6, 2016
Additional Notes:
The first-named author was partially supported by ANR grants 08-BLAN-0317-01/02, 09-JCJC-0102-01, 13-BS01-0001-01, and 13-IS01-0001-01/02
Article copyright:
© Copyright 2016
University Press, Inc.
