logo AFST
The dual boundary complex of the SL 2 character variety of a punctured sphere
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 SL 2 (). On considère la variété de caractères des systèmes locaux sur 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 SL 2 (). Consider the character variety of local systems on 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.

@article{AFST_2016_6_25_2-3_317_0,
     author = {Carlos Simpson},
     title = {The dual boundary complex of the $SL_2$ character variety of a punctured sphere},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {317--361},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {2-3},
     year = {2016},
     doi = {10.5802/afst.1496},
     mrnumber = {3530160},
     zbl = {1352.14009},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1496/}
}
TY  - JOUR
AU  - Carlos Simpson
TI  - The dual boundary complex of the $SL_2$ character variety of a punctured sphere
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2016
DA  - 2016///
SP  - 317
EP  - 361
VL  - Ser. 6, 25
IS  - 2-3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1496/
UR  - https://www.ams.org/mathscinet-getitem?mr=3530160
UR  - https://zbmath.org/?q=an%3A1352.14009
UR  - https://doi.org/10.5802/afst.1496
DO  - 10.5802/afst.1496
LA  - en
ID  - AFST_2016_6_25_2-3_317_0
ER  - 
%0 Journal Article
%A Carlos Simpson
%T The dual boundary complex of the $SL_2$ character variety of a punctured sphere
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2016
%P 317-361
%V Ser. 6, 25
%N 2-3
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1496
%R 10.5802/afst.1496
%G en
%F AFST_2016_6_25_2-3_317_0
Carlos Simpson. The dual boundary complex of the $SL_2$ character variety of a punctured sphere. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 317-361. doi : 10.5802/afst.1496. https://afst.centre-mersenne.org/articles/10.5802/afst.1496/

[1] Abramovich (D.), Corti (A.), and Vistoli (A.).— Twisted bundles and admissible covers. Comm. Algebra 31, p. 3547-3618 (2003). | Article | MR 2007376 | Zbl 1077.14034

[2] Berkovich (V. G.).— Spectral Theory and Analytic Geometry over non-Archimedean fields. Mathematical Surveys and Monographs 33, AMS, Providence (1990). | Zbl 0715.14013

[3] D’Adderio (M.), Moci (L.).— (2013). Arithmetic matroids, the Tutte polynomial and toric arrangements. Advances in Math. 232, p. 335-367 (2013). | Article | MR 2989987 | Zbl 1256.05039

[4] Danilov (V.).— Polyhedra of schemes and algebraic varieties. Mathematics of the USSR-Sbornik 26, p. 137-149 (1975). | Article | MR 441970

[5] Daskalopoulos (G.), Dostoglou (S.), Wentworth (R.).— On the Morgan-Shalen compactification of the SL(2,) character varieties of surface groups. Duke Math. J. 101, p. 189-207 (2000). | Article | Zbl 0974.58009

[6] Davison (B.).— Cohomological Hall algebras and character varieties. Preprint arXiv:1504.00352 (2015). | Article | MR 3521588 | Zbl 1348.14043

[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). | Article | MR 2912707 | Zbl 1375.14047

[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 1109.14307

[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). | Article | MR 2891861

[12] Frenkel (E.), Ben-Zvi (D.).— Vertex algebras and algebraic curves. Mathematical Surveys and Monographs 88, AMS, Providence (2001). | Zbl 0981.17022

[13] Gaiotto (D.), Moore (G.W.), Neitzke (A.).— Spectral networks. Annales Henri Poincaré 14, p. 1643-1731 (2013). | Article | MR 3115984 | Zbl 1288.81132

[14] Godinho (L.), Mandini (A.).— Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles. Advances in Mathematics 244, p. 465-532 (2013). | Article | MR 3077880 | Zbl 1318.32018

[15] Goldman (W.).— The complex-symplectic geometry of SL(2,)-characters over surfaces. Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, p. 375-407 (2004). | Zbl 1089.53060

[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). | Article | MR 3415066 | Zbl 1351.14024

[17] Gross (M.), Hacking (P.), Keel (S.), Kontsevich (M.).— Canonical bases for cluster algebras. Preprint arXiv:1411.1394 (2014). | Article | MR 3758151

[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 1322.14027

[19] Hausel (T.), Letellier (E.), Rodriguez-Villegas (F.).— Arithmetic harmonic analysis on character and quiver varieties, Duke Math. J. 160 p. 323-400 (2011). | Article | MR 2852119 | Zbl 1246.14063

[20] Hausel (T.), Letellier (E.), Rodriguez-Villegas (F.).— Arithmetic harmonic analysis on character and quiver varieties II, Adv. Math. 234, p. 85-128 (2013). | Article | MR 3003926 | Zbl 1273.14101

[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). | Article | MR 1949162 | Zbl 1015.14018

[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). | Article | MR 2044052 | Zbl 1060.14048

[23] Hausel (T.), Rodriguez-Villegas (F.).— Mixed Hodge polynomials of character varieties. Invent. Math. 174, p. 555-624 (2008). | Article | MR 2453601 | Zbl 1213.14020

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

[25] Hitchin (N.).— Stable bundles and integrable systems. Duke Math. J. 54, p. 91-114 (1987). | Article | MR 885778 | Zbl 0627.14024

[26] Hitchin (N.).— The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, p. 59-126 (1987). | Article | MR 887284 | Zbl 0634.53045

[27] Hollands (L.), Neitzke (A.).— Spectral networks and Fenchel-Nielsen coordinates. Preprint arXiv:1312.2979 (2013). | Article | MR 3500424

[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). | Article | MR 1427677 | Zbl 0911.58017

[29] Kabaya (Y.).— Parametrization of PSL (2,)-representations of surface groups. Geometriae Dedicata 170, p. 9-62 (2014). | Article | MR 3199475 | Zbl 1290.30054

[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). | Article | MR 3322389 | Zbl 1314.32021

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

[32] Kollár (J.), Xu (C.).— The dual complex of Calabi-Yau pairs. Preprint arXiv:1503.08320 (2015). | Article | MR 3539921

[33] Komyo (A.).— On compactifications of character varieties of n-punctured projective line. Preprint arXiv:1307.7880 (2013). | Article | MR 3449188

[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). | Article | MR 1206652 | Zbl 0787.53022

[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). | Article

[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). | Article

[37] Kostov (V.).— On the Deligne-Simpson problem. Proc. Steklov Inst. Math. 238, p. 148-185 (2002). | Zbl 1036.34106

[38] Letellier (E.).— Character varieties with Zariski closures of GL n -conjugacy classes at punctures. Selecta 21, p. 293-344 (2015). | Article | MR 3300418 | Zbl 1400.14122

[39] Manon (C.).— Toric geometry of SL 2 () free group character varieties from outer space. Preprint arXiv:1410.0072 (2014). | Article

[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 0881.14006

[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). | Article | MR 2851597

[42] Nicaise (J.), Xu (C.).— The essential skeleton of a degeneration of algebraic varieties. Preprint arXiv:1307.4041 (2013). | Article | MR 3595497

[43] Noohi (B.).— Fundamental groups of algebraic stacks. J. Inst. Math. Jussieu 3, p. 69-103 (2004). | Article | MR 2036598

[44] Parker (J.), Platis (I.).— Complex hyperbolic Fenchel-Nielsen coordinates. Topology 47, p. 101-135 (2008). | Article | MR 2415771 | Zbl 1169.30019

[45] Payne (S.).— Boundary complexes and weight filtrations. Michigan Math. J. 62, p. 293-322 (2013). | Article | MR 3079265 | Zbl 1312.14049

[46] Simpson (C.).— Local systems on proper algebraic V-manifolds. Pure and Appl. Math. Quarterly (Eckart Viehweg’s volume), 7, p. 1675-1760 (2011). | Article | MR 2918179 | Zbl 1316.14008

[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). | Article | MR 3471271

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

[49] Stepanov (D.A.).— A note on resolution of rational and hypersurface singularities. Proc. Amer. Math. Soc. 136, p. 2647-2654 (2008). | Article | MR 2399025 | Zbl 1144.14002

[50] Tan (S.P.).— Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures. International J. Math. 5, p. 239-251 (1994). | Article | MR 1266284 | Zbl 0816.32017

[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). | Article | Zbl 1134.14018

[52] Schechtman (V.), Varchenko (A.).— Hypergeometric solutions of Knizhnik-Zamolodchikov equations. Lett. Math. Phys. 20, p. 279-283 (1990). | Article | MR 1077959

[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). | Article | MR 1480881 | Zbl 0919.14018

[54] Wolpert (S.).— The Fenchel-Nielsen deformation. Annals of Math., p. 501-528 (1982). | Article | MR 657237 | Zbl 0496.30039

Cité par Sources :