The dual boundary complex of the $S{L}_2$ character variety of a punctured sphere

Simpson, Carlos

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 317-361.

Soient $C_1,...,C_k$ des classes de conjugaison génériques dans $S{L}_2\left(ℂ\right)$. On considère la variété de caractères des systèmes locaux sur ${\mathbb{P}}^1-\{y_1,...,y_k\}$ dont les transformations de monodromie autour des $y_i$ sont dans les classes de conjugaison $C_i$ respectives. On montre que le complexe dual du bord de cette variété est équivalent par homotopie à un sphère de dimension $2(k-3)-1.$

Suppose $C_1,...,C_k$ are generic conjugacy classes in $S{L}_2\left(ℂ\right)$. Consider the character variety of local systems on ${\mathbb{P}}^1-\{y_1,...,y_k\}$ whose monodromy transformations around the punctures $y_i$ are in the respective conjugacy classes $C_i$. We show that the dual boundary complex of this character variety is homotopy equivalent to a sphere of dimension $2(k-3)-1.$

[1] Abramovich (D.), Corti (A.), and Vistoli (A.).— Twisted bundles and admissible covers. Comm. Algebra 31, p. 3547-3618 (2003). | DOI | MR | Zbl

[2] Berkovich (V. G.).— Spectral Theory and Analytic Geometry over non-Archimedean fields. Mathematical Surveys and Monographs 33, AMS, Providence (1990). | Zbl

[3] D’Adderio (M.), Moci (L.).— (2013). Arithmetic matroids, the Tutte polynomial and toric arrangements. Advances in Math. 232, p. 335-367 (2013). | DOI | MR | Zbl

[4] Danilov (V.).— Polyhedra of schemes and algebraic varieties. Mathematics of the USSR-Sbornik 26, p. 137-149 (1975). | DOI | MR

[5] Daskalopoulos (G.), Dostoglou (S.), Wentworth (R.).— On the Morgan-Shalen compactification of the $SL(2,ℂ)$ character varieties of surface groups. Duke Math. J. 101, p. 189-207 (2000). | DOI | Zbl

[6] Davison (B.).— Cohomological Hall algebras and character varieties. Preprint arXiv:1504.00352 (2015). | DOI | MR | Zbl

[7] de Cataldo (M. A.), Hausel (T.), Migliorini (L.).— Topology of Hitchin systems and Hodge theory of character varieties: the case A1. Ann. of Math. 175, p. 1329-1407 (2012). | DOI | MR | Zbl

[8] de Fernex (T.), Kollár (J.), Xu (C.).— The dual complex of singularities. Preprint arXiv:1212.1675 (2012).

[9] Drézet (J.-M.).— Luna’s slice theorem and applications. Algebraic group actions and quotients, J. A. Wisniewski, ed., Hindawi, p. 39-90 (2004). | Zbl

[10] Fenchel (W.), Nielsen (J.).— Discontinuous groups of non-Euclidean motions. Unpublished manuscript.

[11] Francis (J.), Gaitsgory (D.).— Chiral Koszul duality. Selecta Mathematica 18, p. 27-87 (2012). | DOI | MR

[12] Frenkel (E.), Ben-Zvi (D.).— Vertex algebras and algebraic curves. Mathematical Surveys and Monographs 88, AMS, Providence (2001). | Zbl

[13] Gaiotto (D.), Moore (G.W.), Neitzke (A.).— Spectral networks. Annales Henri Poincaré 14, p. 1643-1731 (2013). | DOI | MR | Zbl

[14] Godinho (L.), Mandini (A.).— Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles. Advances in Mathematics 244, p. 465-532 (2013). | DOI | MR | Zbl

[15] Goldman (W.).— The complex-symplectic geometry of $SL(2,ℂ)$-characters over surfaces. Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, p. 375-407 (2004). | Zbl

[16] Gross (M.), Hacking (P.), Keel (S.).— Mirror symmetry for log Calabi-Yau surfaces I. Publ. Math. I.H.E.S. 122, p. 65-168 (2015). | DOI | MR | Zbl

[17] Gross (M.), Hacking (P.), Keel (S.), Kontsevich (M.).— Canonical bases for cluster algebras. Preprint arXiv:1411.1394 (2014). | DOI | MR

[18] Hausel (T.).— Global topology of the Hitchin system. Handbook of moduli, Vol. II, Adv. Lect. Math. (ALM) 25, Int. Press, p. 29-69 (2013). | Zbl

[19] Hausel (T.), Letellier (E.), Rodriguez-Villegas (F.).— Arithmetic harmonic analysis on character and quiver varieties, Duke Math. J. 160 p. 323-400 (2011). | DOI | MR | Zbl

[20] Hausel (T.), Letellier (E.), Rodriguez-Villegas (F.).— Arithmetic harmonic analysis on character and quiver varieties II, Adv. Math. 234, p. 85-128 (2013). | DOI | MR | Zbl

[21] Hausel (T.), Thaddeus (M.).— Relations in the cohomology ring of the moduli space of rank $2$ Higgs bundles. J.A.M.S. 16, p. 303-329 (2003). | DOI | MR | Zbl

[22] Hausel (T.), Thaddeus (M.).— Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles. Proc. London Math. Soc. 88, p. 632-658 (2004). | DOI | MR | Zbl

[23] Hausel (T.), Rodriguez-Villegas (F.).— Mixed Hodge polynomials of character varieties. Invent. Math. 174, p. 555-624 (2008). | DOI | MR | Zbl

[24] Hinich (V.), Schechtman (V.).— On homotopy limit of homotopy algebras. K-theory, Arithmetic and Geometry, Springer, p. 240-264 (1987). | DOI

[25] Hitchin (N.).— Stable bundles and integrable systems. Duke Math. J. 54, p. 91-114 (1987). | DOI | MR | Zbl

[26] Hitchin (N.).— The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, p. 59-126 (1987). | DOI | MR | Zbl

[27] Hollands (L.), Neitzke (A.).— Spectral networks and Fenchel-Nielsen coordinates. Preprint arXiv:1312.2979 (2013). | DOI | MR

[28] Jeffrey (L.), Weitsman (J.).— Toric structures on the moduli space of flat connections on a Riemann surface II: Inductive decomposition of the moduli space. Math. Annalen 307, p. 93-108 (1997). | DOI | MR | Zbl

[29] Kabaya (Y.).— Parametrization of $\mathrm{PSL}(2,ℂ)$-representations of surface groups. Geometriae Dedicata 170, p. 9-62 (2014). | DOI | MR | Zbl

[30] Katzarkov (L.), Noll (A.), Pandit (P.), Simpson (C.).— Harmonic maps to buildings and singular perturbation theory. Comm. Math. Physics 336, p. 853-903 (2015). | DOI | MR | Zbl

[31] Katzarkov (L.), Noll (A.), Pandit (P.), Simpson (C.).— Constructing buildings and harmonic maps. Preprint arXiv:1503.00989 (2015). | DOI

[32] Kollár (J.), Xu (C.).— The dual complex of Calabi-Yau pairs. Preprint arXiv:1503.08320 (2015). | DOI | MR

[33] Komyo (A.).— On compactifications of character varieties of $n$-punctured projective line. Preprint arXiv:1307.7880 (2013). | DOI | MR

[34] Konno (H.).— Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface. J. Math. Soc. Japan 45, p. 253-276 (1993). | DOI | MR | Zbl

[35] Kontsevich (M.), Soibelman (Y.).— Homological mirror symmetry and torus fibrations. Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publishing, p. 203-263 (2001). | DOI

[36] Kontsevich (M.), Soibelman (Y.).— Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry. Homological Mirror Symmetry and Tropical Geometry, R. Castano-Bernard et al eds., Springer, p. 197-308 (2014). | DOI

[37] Kostov (V.).— On the Deligne-Simpson problem. Proc. Steklov Inst. Math. 238, p. 148-185 (2002). | Zbl

[38] Letellier (E.).— Character varieties with Zariski closures of $G{L}_n$-conjugacy classes at punctures. Selecta 21, p. 293-344 (2015). | DOI | MR | Zbl

[39] Manon (C.).— Toric geometry of $S{L}_2\left(ℂ\right)$ free group character varieties from outer space. Preprint arXiv:1410.0072 (2014). | DOI

[40] Nakajima (H.).— Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces. Moduli of vector bundles (Sanda, Kyoto 1994), M. Maruyama ed., Lecture notes in pure and applied math., p. 199-208 (1996). | Zbl

[41] Nekrasov (N.), Rosly (A.), Shatashvili (S.).— (2011). Darboux coordinates, Yang-Yang functional, and gauge theory. Nuclear Physics B-Proceedings Supplements 216, p. 69-93 (2011). | DOI | MR

[42] Nicaise (J.), Xu (C.).— The essential skeleton of a degeneration of algebraic varieties. Preprint arXiv:1307.4041 (2013). | DOI | MR

[43] Noohi (B.).— Fundamental groups of algebraic stacks. J. Inst. Math. Jussieu 3, p. 69-103 (2004). | DOI | MR

[44] Parker (J.), Platis (I.).— Complex hyperbolic Fenchel-Nielsen coordinates. Topology 47, p. 101-135 (2008). | DOI | MR | Zbl

[45] Payne (S.).— Boundary complexes and weight filtrations. Michigan Math. J. 62, p. 293-322 (2013). | DOI | MR | Zbl

[46] Simpson (C.).— Local systems on proper algebraic $V$-manifolds. Pure and Appl. Math. Quarterly (Eckart Viehweg’s volume), 7, p. 1675-1760 (2011). | DOI | MR | Zbl

[47] Soibelman (A.).— The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem. Preprint arXiv:1310.1144 (2013). | DOI | MR

[48] Stepanov (D.A.).— A remark on the dual complex of a resolution of singularities. Uspekhi Mat. Nauk 61 (367), p. 185-186 (2006). | DOI

[49] Stepanov (D.A.).— A note on resolution of rational and hypersurface singularities. Proc. Amer. Math. Soc. 136, p. 2647-2654 (2008). | DOI | MR | Zbl

[50] Tan (S.P.).— Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures. International J. Math. 5, p. 239-251 (1994). | DOI | MR | Zbl

[51] Thuillier (A.).— Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math. 123, p. 381-451 (2007). | DOI | Zbl

[52] Schechtman (V.), Varchenko (A.).— Hypergeometric solutions of Knizhnik-Zamolodchikov equations. Lett. Math. Phys. 20, p. 279-283 (1990). | DOI | MR

[53] Weitsman (J.).— Geometry of the intersection ring of the moduli space of flat connections and the conjectures of Newstead and Witten. Topology 37, p. 115-132 (1998). | DOI | MR | Zbl

[54] Wolpert (S.).— The Fenchel-Nielsen deformation. Annals of Math., p. 501-528 (1982). | DOI | MR | Zbl