Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T05:28:23.112Z Has data issue: false hasContentIssue false
  • English
  • Français

Goldman Systems and Bending Systems

Published online by Cambridge University Press:  20 November 2018

Yuichi Nohara  and
Kazushi Ueda
Yuichi Nohara
Affiliation:
Faculty of Education, Kagawa University, Saiwai-cho 1-1, Takamatsu, Kagawa, 760-8522, Japan. [email protected]
Kazushi Ueda
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan. [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and as symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.

Keywords

53D3014H60toric degeneration
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 5 , 01 October 2015 , pp. 1109 - 1143
Copyright
Copyright © Canadian Mathematical Society 2015

References

[AMM98] Alekseev, A., Malkin, A., and Meinrenken, E., Lie group valued moment maps. J. Differential Geom. 48(1998), no. 3, 445–495.Google Scholar
[Bau91] Bauer, S., Parabolic bundles, elliptic surfaces and S\J {2)–representation spaces of genus zero Fuchsian groups. Math. Ann. 290(1991), no. 3, 509–526.http://dx.doi.org/10.1007/BF01459257 Google Scholar
[DH98] Dolgachev, I. V. and Hu, Y., Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. 87(1998), 5–56.Google Scholar
[FH05] Foth, P. and Hu, Y., Toric degenerations of weight varieties and applications. In: Travaux mathématiques. Fasc. XVI, Trav. Math., XVI, Univ. Luxemb., Luxembourg, 2005, pp. 87–105.Google Scholar
[GHJW97] Guruprasad, K., Huebschmann, J., Jeffrey, L., and Weinstein, A., Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89 (1997), no. 2, 377–412.http://dx.doi.org/10.1215/S0012-7094-97-08917-1 Google Scholar
[GL87] Geigle, W. and Lenzing, H., A class of weighted protective curves arising in representation theory of finite–dimensional algebras. In: Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lecture Notes in Math., 1273, Springer, Berlin, 1987, pp. 265–297.Google Scholar
[Gol86] Goldman, W. M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85(1986), no. 2, 263–302.http://dx.doi.Org/10.1007/BF01389091 Google Scholar
[GS84] Guillemin, V. and Sternberg, S., A normal form for the moment map. In: Differential geometric methods in mathematical physics (Jerusalem, 1982), Math. Phys. Stud., 6, Reidel, Dordrecht, 1984, pp. 161–175.Google Scholar
[HJOO] Hurtubise, J. C. and Jeffrey, L. C., Representations with weighted frames and framed parabolic bundles. Canad. J. Math. 52(2000), no. 6,1235–1268.http://dx.doi.org/10.4153/CJM–2000–052–4 Google Scholar
[HK97] Hausmann, J.-C. and Knutson, A., Polygon spaces and Grassmannians. Enseign. Math. (2) 43(1997), no. 1–2, 173–198.Google Scholar
[HMM11] Howard, B., Manon, C., and Millson, J., The toric geometry of triangulated polygons in Euclidean space. Canad. J. Math. 63(2011), no. 4, 878–937.http://dx.doi.Org/10.4153/CJM-2O11-021-0 Google Scholar
[Jef94] Jeffrey, L. C., Extended moduli spaces of flat connections on Riemann surfaces. Math. Ann. 298(1994), no. 4, 667–692.http://dx.doi.org/10.1007/BF01459756 Google Scholar
[JW92] Jeffrey, L. C. and Weitsman, J., Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Comm. Math. Phys. 150(1992), no. 3, 593–630.http://dx.doi.org/10.1007/BF02096964 Google Scholar
[JW94] Jeffrey, L. C. and Weitsman, J., Toric structures on the moduli space of flat connections on a Riemann surface: volumes and the moment map. Adv. Math. 106(1994), no. 2, 151–168.http://dx.doi.Org/10.1006/aima.1994.1054 Google Scholar
[JW97] Jeffrey, L. C. and Weitsman, J., Toric structures on the moduli space of flat connections on a Riemann surface. II. Inductive decomposition of the moduli space. Math. Ann. 307(1997), no. 1, 93–108.http://dx.doi.Org/10.1OO7/sOO2O8OO5OO24 Google Scholar
[Kly94] Klyachko, A. A., Spatial polygons and stable configurations of points in the projective line. In: Algebraic geometry and its applications (Yaroslavl, 1992), Aspects Math., E25, Vieweg, Braunschweig, 1994, pp. 67–84.Google Scholar
[KM96] Kapovich, M. and Millson, J. J., The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 44(1996), no. 3, 479–513.Google Scholar
[KY02] Kamiyama, Y. and Yoshida, T., Symplectic toric space associated to triangle inequalities. Geom. Dedicata 93(2002), 25–36. http://dx.doi.Org/10.1023/A:1020393910472 Google Scholar
[Lenll] Lenzing, H., Weighted projective lines and applications. In: Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc, Zürich, 2011, pp. 153–187.Google Scholar
[Man] Manon, C. A., The algebra of conformai blocks. arxiv:0910.0577Google Scholar
[Mar85] Marie, C.-M., Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectiaue. Rend. Sem. Mat. Univ. Politec. Torino 43(1985), no. 2, 227–251.Google Scholar
[MP01] Millson, J. J. and Poritz, J. A., Around polygons in ℝ3 and S3. Comm. Math. Phys. 218(2001), no. 2, 315–331.http://dx.doi.Org/10.1007/PL00005557 Google Scholar
[MS80] Mehta, V. B. and Seshadri, C. S., Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248(1980), no. 3, 205–239.http://dx.doi.Org/10.1007/BF01420526 Google Scholar
[NNUIO] Nishinou, T., Nohara, Y., and Ueda, K., Toric degenerations of Gelfand-Cetlin systems and potential functions. Adv. Math. 224(2010), no. 2, 648–706.http://dx.doi.Org/10.1016/j.aim.2009.12.012 Google Scholar
[NU14] Nohara, Y. and Ueda, K., Toric degenerations of integrable systems on Grassmannians and polygon spaces. Nagoya Math. J. 214(2014), 125–168.http://dx.doi.org/10.1215/00277630-2643839 Google Scholar
[SS04] Speyer, D. and Sturmfels, B., The tropical Grassmannian. Adv. Geom. 4(2004), no. 3, 389–411.Google Scholar
[Tha96] Thaddeus, M., Geometric invariant theory and flips. J. Amer. Math. Soc. 9(1996), no. 3, 691–723.http://dx.doi.org/10.1090/S0894-0347-96-00204-4 Google Scholar
[TreO2] Treloar, T., The symplectic geometry of polygons in the 3–sphere. Canad. J. Math. 54(2002),no. 1, 30–54.http://dx.doi.Org/10.4153/CJM-2OO2-OO2-1 Google Scholar
[Wei92] Weitsman, J., Real polarization of the moduli space of flat connections on a Riemann surface. Comm. Math. Phys. 145(1992), no. 3, 425–433.http://dx.doi.org/10.1007/BF02099391 Google Scholar