On compactifications of character varieties of n-punctured projective line
Annales de l'Institut Fourier, Volume 65 (2015) no. 4, p. 1493-1523

In this paper, we construct compactifications of SL 2 ()-character varieties of n-punctured projective line and study the boundary divisors of the compactifications. This study is motivated by a conjecture for the configurations of the boundary divisors, due to C. Simpson. We verify the conjecture for a few examples.

Dans cet article, nous construisons des compactifications de SL 2 ()-variétés de caractères d’une droite projective moins n points et étudions les diviseurs au bord des compactifications. Cette étude est motivée par une conjecture, due à C. Simpson, sur les configurations des diviseurs au bord. Nous vérifions quelques cas de la conjecture.

DOI : https://doi.org/10.5802/aif.2965
Classification:  14L24,  14L30
Keywords: character variety, geometric invariant theory
@article{AIF_2015__65_4_1493_0,
     author = {Komyo, Arata},
     title = {On compactifications of character varieties of $n$-punctured projective line},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {4},
     year = {2015},
     pages = {1493-1523},
     doi = {10.5802/aif.2965},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_4_1493_0}
}
Komyo, Arata. On compactifications of character varieties of $n$-punctured projective line. Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1493-1523. doi : 10.5802/aif.2965. http://www.numdam.org/item/AIF_2015__65_4_1493_0/

[1] De Cataldo, Mark Andrea A.; Hausel, Tamás; Migliorini, Luca Topology of Hitchin systems and Hodge theory of character varieties: the case A 1 , Ann. of Math. (2), Tome 175 (2012) no. 3, pp. 1329-1407 | Article | MR 2912707

[2] Formanek, Edward The invariants of n×n matrices, Invariant theory, Springer, Berlin (Lecture Notes in Math.) Tome 1278 (1987), pp. 18-43 | Article | MR 924163 | Zbl 0645.16012

[3] Fricke, Robert; Klein, Felix Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttg art, Bibliotheca Mathematica Teubneriana, Bände 3, Tome 4 (1965), pp. Band I: xiv+634 pp.; Band II: xiv+668 | MR 183872

[4] Hausel, Tamás; Letellier, Emmanuel; Rodriguez-Villegas, Fernando Arithmetic harmonic analysis on character and quiver varieties, Duke Math. J., Tome 160 (2011) no. 2, pp. 323-400 | Article | MR 2852119 | Zbl 1246.14063

[5] Hausel, Tamás; Rodriguez-Villegas, Fernando Mixed Hodge polynomials of character varieties, Invent. Math., Tome 174 (2008) no. 3, pp. 555-624 (With an appendix by Nicholas M. Katz) | Article | MR 2453601 | Zbl 1213.14020

[6] Inaba, Michi-Aki; Iwasaki, Katsunori; Saito, Masa-Hiko Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci., Tome 42 (2006) no. 4, pp. 987-1089 http://projecteuclid.org/euclid.prims/1166642194 | Article | MR 2289083 | Zbl 1127.34055

[7] Inaba, Michi-Aki; Iwasaki, Katsunori; Saito, Masa-Hiko Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, Moduli spaces and arithmetic geometry, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 45 (2006), pp. 387-432 | MR 2310256 | Zbl 1115.14005

[8] Inaba, Michi-Aki; Saito, Masa-Hiko Moduli of unramified irregular singular parabolic connections on a smooth projective curve (http://arxiv.org/abs/1203.0084) | MR 3079310 | Zbl 1267.14015

[9] Iwasaki, Katsunori An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Comm. Math. Phys., Tome 242 (2003) no. 1-2, pp. 185-219 | Article | MR 2018272 | Zbl 1044.34051

[10] Jimbo, Michio Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci., Tome 18 (1982) no. 3, pp. 1137-1161 | Article | MR 688949 | Zbl 0535.34042

[11] Kirwan, Frances Clare Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2), Tome 122 (1985) no. 1, pp. 41-85 | Article | MR 799252 | Zbl 0592.14011

[12] Lawton, Sean Generators, relations and symmetries in pairs of 3×3 unimodular matrices, J. Algebra, Tome 313 (2007) no. 2, pp. 782-801 | Article | MR 2329569 | Zbl 1119.13004

[13] Martin, Benjamin M. S. Compactifications of a representation variety, J. Group Theory, Tome 14 (2011) no. 6, pp. 947-963 | Article | MR 2855788 | Zbl 1279.20057

[14] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Tome 34 (1994), pp. xiv+292 | Article | MR 1304906 | Zbl 0797.14004

[15] Newstead, P. E. Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Tome 51 (1978), pp. vi+183 | MR 546290 | Zbl 1277.14001

[16] Payne, Sam Boundary complexes and weight filtrations (http://arxiv.org/abs/1109.4286) | MR 3079265

[17] Procesi, C. The invariant theory of n×n matrices, Advances in Math., Tome 19 (1976) no. 3, pp. 306-381 | Article | MR 419491 | Zbl 0331.15021

[18] Simpson, Carlos T. Towards the boundary of the character variety (Reference not found)

[19] Simpson, Carlos T. Harmonic bundles on noncompact curves, J. Amer. Math. Soc., Tome 3 (1990) no. 3, pp. 713-770 | Article | MR 1040197 | Zbl 0713.58012

[20] Simpson, Carlos T. Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. (1994) no. 79, pp. 47-129 | Article | Numdam | MR 1307297 | Zbl 0891.14005

[21] Simpson, Carlos T. Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. (1994) no. 80, p. 5-79 (1995) | Article | Numdam | MR 1320603 | Zbl 0891.14006

[22] Stepanov, D. A. A remark on the dual complex of a resolution of singularities, Uspekhi Mat. Nauk, Tome 61 (2006) no. 1(367), p. 185-186 | Article | MR 2239783 | Zbl 1134.14302

[23] Thuillier, Amaury Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscripta Math., Tome 123 (2007) no. 4, pp. 381-451 | Article | MR 2320738 | Zbl 1134.14018