On compactifications of character varieties of $n$-punctured projective line

Komyo, Arata

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1493-1523.

Dans cet article, nous construisons des compactifications de $S{L}_2\left(ℂ\right)$-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.

In this paper, we construct compactifications of $S{L}_2\left(ℂ\right)$-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.

[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), Volume 175 (2012) no. 3, pp. 1329-1407 | DOI | MR

[2] Formanek, Edward The invariants of $n\times n$ matrices, Invariant theory (Lecture Notes in Math.), Volume 1278, Springer, Berlin, 1987, pp. 18-43 | DOI | MR | Zbl

[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, Bibliotheca Mathematica Teubneriana, Bände 3, 4, Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttg art, 1965, pp. Band I: xiv+634 pp.; Band II: xiv+668 | MR

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

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

[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., Volume 42 (2006) no. 4, pp. 987-1089 http://projecteuclid.org/euclid.prims/1166642194 | DOI | MR | Zbl

[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 (Adv. Stud. Pure Math.), Volume 45, Math. Soc. Japan, Tokyo, 2006, pp. 387-432 | MR | Zbl

[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 | Zbl

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

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

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

[12] Lawton, Sean Generators, relations and symmetries in pairs of $3\times 3$ unimodular matrices, J. Algebra, Volume 313 (2007) no. 2, pp. 782-801 | DOI | MR | Zbl

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

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

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

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

[17] Procesi, C. The invariant theory of $n\times n$ matrices, Advances in Math., Volume 19 (1976) no. 3, pp. 306-381 | DOI | MR | Zbl

[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., Volume 3 (1990) no. 3, pp. 713-770 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

[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) | DOI | Numdam | MR | Zbl

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

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