Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-13T07:41:28.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Global geometry on moduli of local systems for surfaces with boundary

Part of: Low-dimensional topology Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 October 2020

Junho Peter Whang
Junho Peter Whang*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Simons Building Room 2-238A, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA jwhang@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that every coarse moduli space, parametrizing complex special linear rank-2 local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi–Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating series for counts of essential multicurves on the surface.

Keywords

character varietycompactificationlog Calabi–Yausurfacemulticurve

MSC classification

Primary: 14J32: Calabi-Yau manifolds
Secondary: 57M05: Fundamental group, presentations, free differential calculus
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 8 , August 2020 , pp. 1517 - 1559
Copyright
Copyright © The Author(s) 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bruns, W. and Herzog, J., Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993).Google Scholar
Charles, L. and Marché, J., Multicurves and regular functions on the representation variety of a surface in SU(2), Comment. Math. Helv. 87 (2012), 409–431.10.4171/CMH/258CrossRefGoogle Scholar
Corti, A. and Kaloghiros, A.-S., The Sarkisov program for Mori fibred Calabi–Yau pairs, Algebr. Geom. 3 (2016), 370–384.CrossRefGoogle Scholar
Demazure, M., Anneaux gradués normaux, in Introduction à la théorie des singularités, II, Travaux en Cours, vol. 37 (Hermann, Paris, 1988), 35–68.Google Scholar
Drensky, V. and Formanek, E., Polynomial identity rings, Advanced Courses in Mathematics – CRM Barcelona (Birkhäuser, Basel, 2004).CrossRefGoogle Scholar
Faltings, G., Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), 549–576.CrossRefGoogle Scholar
Fock, V. and Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211.10.1007/s10240-006-0039-4CrossRefGoogle Scholar
Goldman, W. M., Mapping class group dynamics on surface group representations, in Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74 (American Mathematical Society, Providence, RI, 2006), 189–214.CrossRefGoogle Scholar
Goldman, W. M., Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, in Handbook of Teichmüller theory, vol. II, IRMA Lectures in Mathematics and Theoretical Physics, vol. 13 (European Mathematical Society, Zürich, 2009), 611–684.CrossRefGoogle Scholar
Gross, M., Hacking, P. and Keel, S., Birational geometry of cluster algebras, Algebr. Geom. 2 (2015), 137–175.CrossRefGoogle Scholar
Gross, M., Hacking, P., Keel, S. and Kontsevich, M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608.CrossRefGoogle Scholar
Hochster, M. and Roberts, J. L., Rings of invariants of reductive groups acting on regular rings are Cohen–Macaulay, Adv. Math. 13 (1974), 115–175.Google Scholar
Horowitz, R. D., Characters of free groups represented in the two-dimensional special linear group, Commun. Pure Appl. Math. 25 (1972), 635–649.CrossRefGoogle Scholar
Kollár, J. and Xu, C., The dual complex of Calabi–Yau pairs, Invent. Math. 205 (2016), 527–557.CrossRefGoogle Scholar
Kolláar, J., Conic bundles that are not birational to numerical Calabi–Yau pairs, Preprint (2016).Google Scholar
Komyo, A., On compactifications of character varieties of n-punctured projective line, Ann. Inst. Fourier (Grenoble) 65 (2015), 1493–1523.CrossRefGoogle Scholar
Le Bruyn, L., The functional equation for Poincaré series of trace rings of generic $2\times 2$ matrices, Israel J. Math. 52 (1985), 355–360.Google Scholar
Lee, C. W., The associahedron and triangulations of the $n$-gon, European J. Combin. 10 (1989), 551–560.CrossRefGoogle Scholar
Manon, C., Compactifications of character varieties and skein relations on conformal blocks, Geom. Dedicata 179 (2015), 335–376.Google Scholar
Manon, C., Toric geometry of $\operatorname{SL}_2({\mathbb {C}})$ free group character varieties from outer space, Canad. J. Math. 70 (2018), 354–399.Google Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1986), translated from the Japanese by Reid, M..Google Scholar
Porti, J., Reidemeister torsion, hyperbolic three-manifolds, and character varieties (English summary), in Handbook of group actions, vol. IV, Advanced Lectures in Mathematics (ALM), vol. 41 (International Press, Somerville, MA, 2018), 447–507.Google Scholar
Procesi, C., The invariant theory of $n\times n$ matrices, Adv. Math. 19 (1976), 306–381.CrossRefGoogle Scholar
Przytycki, J. H. and Sikora, A. S., On skein algebras and $\operatorname{SL}_2({\textbf {C}})$-character varieties, Topology 39 (2000), 115–148.CrossRefGoogle Scholar
Saito, K., Character variety of representations of a finitely generated group in SL2, in Topology and Teichmüller spaces (Katinkulta, 1995) (World Scientific, River Edge, NJ, 1996), 253–264.Google Scholar
Simpson, C., The dual boundary complex of the SL2 character variety of a punctured sphere, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016), 317–361.CrossRefGoogle Scholar
Stanley, R. P., Hilbert functions of graded algebras, Adv. Math. 28 (1978), 57–83.Google Scholar
Vogt, H., Sur les invariants fondamentaux des équations différentielles linéaires du second ordre, Ann. Sci. École Norm. Sup. (3) 6 (1889), 3–71.CrossRefGoogle Scholar
Vojta, P., Siegel's theorem in the compact case, Ann. of Math. (2) 133 (1991), 509–548.CrossRefGoogle Scholar
Vojta, P., Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), 133–181.CrossRefGoogle Scholar
Watanabe, K., Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203–211.CrossRefGoogle Scholar
Whang, J. P., Arithmetic of curves on moduli of local systems, Algebra Number Theory, to appear. Preprint (2020), arXiv:1803.04583.Google Scholar
Whang, J. P., Nonlinear descent on moduli of local systems, Israel J. Math., to appear. Preprint (2020), arXiv:1710.01848.Google Scholar