The monodromy of meromorphic projective structures
HTML articles powered by AMS MathViewer
- by Dylan G. L. Allegretti and Tom Bridgeland PDF
- Trans. Amer. Math. Soc. 373 (2020), 6321-6367 Request permission
Abstract:
We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\operatorname {PGL}_2(\mathbb {C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map.References
- Dylan G. L. Allegretti, Stability conditions and cluster varieties from quivers of type $A$, Adv. Math. 337 (2018), 260–293. MR 3853051, DOI 10.1016/j.aim.2018.08.017
- Dylan G. L. Allegretti, Voros symbols as cluster coordinates, J. Topol. 12 (2019), no. 4, 1031–1068. MR 3977870, DOI 10.1112/topo.12106
- D. G. L. Allegretti, Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces, arXiv:1912.05938 [math.AG], 2019.
- Ivar Bakken, A multiparameter eigenvalue problem in the complex plane, Amer. J. Math. 99 (1977), no. 5, 1015–1044. MR 508244, DOI 10.2307/2373998
- W. Balser, W. B. Jurkat, and D. A. Lutz, Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), no. 1, 48–94. MR 545861, DOI 10.1016/0022-247X(79)90217-8
- A. Beilinson and V. Drinfeld, Opers, arXiv:math/0501398 [math.AG], 2005.
- Marco Bertola, Dmitry Korotkin, and Chaya Norton, Symplectic geometry of the moduli space of projective structures in homological coordinates, Invent. Math. 210 (2017), no. 3, 759–814. MR 3735629, DOI 10.1007/s00222-017-0739-z
- Philip P. Boalch, $G$-bundles, isomonodromy, and quantum Weyl groups, Int. Math. Res. Not. 22 (2002), 1129–1166. MR 1904670, DOI 10.1155/S1073792802111081
- P. P. Boalch, Geometry and braiding of Stokes data; fission and wild character varieties, Ann. of Math. (2) 179 (2014), no. 1, 301–365. MR 3126570, DOI 10.4007/annals.2014.179.1.5
- P. P. Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [math.AG], 2015.
- A. A. Bolibruch, S. Malek, and C. Mitschi, On the generalized Riemann-Hilbert problem with irregular singularities, Expo. Math. 24 (2006), no. 3, 235–272. MR 2250948, DOI 10.1016/j.exmath.2005.11.003
- Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI 10.4007/annals.2007.166.317
- Tom Bridgeland, Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math. 216 (2019), no. 1, 69–124. MR 3935038, DOI 10.1007/s00222-018-0843-8
- Tom Bridgeland and Ivan Smith, Quadratic differentials as stability conditions, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 155–278. MR 3349833, DOI 10.1007/s10240-014-0066-5
- Leonid O. Chekhov, Marta Mazzocco, and Vladimir N. Rubtsov, Painlevé monodromy manifolds, decorated character varieties, and cluster algebras, Int. Math. Res. Not. IMRN 24 (2017), 7639–7691. MR 3802126, DOI 10.1093/imrn/rnw219
- É. Delabaere, H. Dillinger, and F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1, 163–199 (French, with English and French summaries). MR 1209700, DOI 10.5802/aif.1326
- David Dumas, Complex projective structures, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 455–508. MR 2497780, DOI 10.4171/055-1/13
- Clifford J. Earle, On variation of projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 87–99. MR 624807
- Gerd Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (1983), no. 2, 223–269. MR 700005
- Vladimir Fock and Alexander Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211. MR 2233852, DOI 10.1007/s10240-006-0039-4
- Sergey Fomin, Michael Shapiro, and Dylan Thurston, Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), no. 1, 83–146. MR 2448067, DOI 10.1007/s11511-008-0030-7
- Edward Frenkel, Langlands correspondence for loop groups, Cambridge Studies in Advanced Mathematics, vol. 103, Cambridge University Press, Cambridge, 2007. MR 2332156
- Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), no. 1, 163–224. MR 2672801, DOI 10.1007/s00220-010-1071-2
- Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239–403. MR 3003931, DOI 10.1016/j.aim.2012.09.027
- Daniel Gallo, Michael Kapovich, and Albert Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2) 151 (2000), no. 2, 625–704. MR 1765706, DOI 10.2307/121044
- R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR 0207977
- Dennis A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), no. 1, 1–55. MR 463429, DOI 10.1007/BF02392015
- John H. Hubbard, The monodromy of projective structures, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 257–275. MR 624819
- Kohei Iwaki and Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014), no. 47, 474009, 98. MR 3280000, DOI 10.1088/1751-8113/47/47/474009
- Katsunori Iwasaki, Moduli and deformation for Fuchsian projective connections on a Riemann surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 (1991), no. 3, 431–531. MR 1146359
- Katsunori Iwasaki, Fuchsian moduli on a Riemann surface—its Poisson structure and Poincaré-Lefschetz duality, Pacific J. Math. 155 (1992), no. 2, 319–340. MR 1178029, DOI 10.2140/pjm.1992.155.319
- Shingo Kawai, The symplectic nature of the space of projective connections on Riemann surfaces, Math. Ann. 305 (1996), no. 1, 161–182. MR 1386110, DOI 10.1007/BF01444216
- T. Koike and R. Schäfke, On the Borel summability of WKB solutions of Schrödinger equations with polynomial potentials and its application, to appear.
- D. Korotkin, Periods of meromorphic quadratic differentials and Goldman bracket, Topological recursion and its influence in analysis, geometry, and topology, Proc. Sympos. Pure Math., vol. 100, Amer. Math. Soc., Providence, RI, 2018, pp. 491–515. MR 3888789
- J. Le Potier, Lectures on vector bundles, Cambridge Studies in Advanced Mathematics, vol. 54, Cambridge University Press, Cambridge, 1997. Translated by A. Maciocia. MR 1428426
- Frank Loray and David Marín Pérez, Projective structures and projective bundles over compact Riemann surfaces, Astérisque 323 (2009), 223–252 (English, with English and French summaries). MR 2647972
- Feng Luo, Monodromy groups of projective structures on punctured surfaces, Invent. Math. 111 (1993), no. 3, 541–555. MR 1202134, DOI 10.1007/BF01231297
- Rolf Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math. 58 (1932), no. 1, 295–373 (German). MR 1555350, DOI 10.1007/BF02547780
- H. Poincaré, Sur les groupes des équations linéaires, Acta Math. 4 (1884), no. 1, 201–312 (French). MR 1554639, DOI 10.1007/BF02418420
- Claude Sabbah, Isomonodromic deformations and Frobenius manifolds, Translated from the 2002 French edition, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2007. An introduction. MR 2368364
- Yasutaka Sibuya, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, Vol. 18, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0486867
- Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
- A. N. Tjurin, Periods of quadratic differentials, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 149–195, 272 (Russian). MR 526014
Additional Information
- Dylan G. L. Allegretti
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
- MR Author ID: 1189385
- Tom Bridgeland
- Affiliation: School of Mathematics and Statistics, University of Sheffield, Western Bank, Sheffield, S10 2TN United Kingdom
- MR Author ID: 635821
- ORCID: 0000-0001-5120-006X
- Received by editor(s): March 18, 2019
- Received by editor(s) in revised form: January 1, 2020
- Published electronically: June 24, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 6321-6367
- MSC (2010): Primary 30F30, 34M40, 57M50; Secondary 13F60, 18E30, 34M60
- DOI: https://doi.org/10.1090/tran/8093
- MathSciNet review: 4155179