Intersection cohomology of character varieties for punctured Riemann surfaces

Mathieu Ballandras

doi:10.5802/jep.215

Intersection cohomology of character varieties for punctured Riemann surfaces
[Cohomologie d’intersection des variétés de caractères des surfaces de Riemann épointées]

Mathieu Ballandras ¹

¹ ICMAT, Campus Cantoblanco, UAM C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 141-198.

Résumé
Abstract

Nous étudions la cohomologie d’intersection des variétés de caractères des surfaces de Riemann épointées, la monodromie autour des points enlevés étant fixée. En nous appuyant sur un résultat de Mellit [Mel20a] pour des monodromies semi-simples, nous calculons la cohomologie d’intersection des variétés de caractères avec des monodromies ayant un type de Jordan quelconque. Ceci prouve la spécialisation au polynôme de Poincaré d’une conjecture de Letellier [Let15].

We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on a previous result from Mellit [Mel20a] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let15].

Reçu le : 2022-02-25
Accepté le : 2022-12-16
Publié le : 2023-01-12

DOI : 10.5802/jep.215

Classification : 14M35, 14F43
Keywords: Character varieties, intersection cohomology, parabolic Higgs bundles
Mot clés : Variétés de caractères, cohomologie d’intersection, fibrés de Higgs paraboliques

Affiliations des auteurs :

Mathieu Ballandras ¹

¹ ICMAT, Campus Cantoblanco, UAM C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Mathieu Ballandras. Intersection cohomology of character varieties for punctured Riemann surfaces. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 141-198. doi : 10.5802/jep.215. https://jep.centre-mersenne.org/articles/10.5802/jep.215/

Bibliographie
Cité par

[AB83] M. F. Atiyah & R. Bott - “The Yang-Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) no. 1505, p. 523-615 | MR | Zbl

[AHS78] M. F. Atiyah, N. J. Hitchin & I. M. Singer - “Self-duality in four-dimensional Riemannian geometry”, Proc. Roy. Soc. London Ser. A 362 (1978) no. 1711, p. 425-461 | MR | Zbl

[BB04] O. Biquard & P. Boalch - “Wild non-abelian Hodge theory on curves”, Compositio Math. 140 (2004) no. 1, p. 179–204 | DOI | MR | Zbl

[BBDG82] A. A. Beilinson, J. Bernstein, P. Deligne & O. Gabber - Faisceaux pervers, Astérisque, vol. 100, Société Mathématique de France, Paris, 1982, ed. 2018

[BD12] I. Biswas & A. Dey - “SYZ duality for parabolic Higgs moduli spaces”, Nuclear Phys. B 862 (2012) no. 1, p. 327-340 | DOI | MR | Zbl

[BGM20] O. Biquard, O. García-Prada & I. Mundet i Riera - “Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group”, Adv. Math. 372 (2020), article ID 107305, 70 pages | DOI | MR | Zbl

[BH17] D. Baraglia & P. Hekmati - “Arithmetic of singular character varieties and their E-polynomials”, Proc. London Math. Soc. (3) 114 (2017) no. 2, p. 293-332 | DOI | MR | Zbl

[Biq97] O. Biquard - “Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse)”, Ann. Sci. École Norm. Sup. (4) 30 (1997) no. 1, p. 41-96 | DOI | Numdam | Zbl

[BM83] W. Borho & R. MacPherson - “Partial resolutions of nilpotent varieties”, in Analysis and topology on singular spaces, II, III, Astérisque, vol. 101, Société Mathématique de France, Paris, 1983 | Numdam | MR | Zbl

[BP16] P. Boalch & R. Paluba - “Symmetric cubic surfaces and $G_{2}$ character varieties”, J. Algebraic Geom. 25 (2016) no. 4, p. 607-631 | DOI | MR | Zbl

[BW79] J. W. Bruce & C. T. C. Wall - “On the classification of cubic surfaces”, J. London Math. Soc. (2) 19 (1979) no. 2, p. 245-256 | DOI | MR | Zbl

[BY96] H. Boden & K. Yokogawa - “Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I”, Internat. J. Math. 7 (1996) no. 5, p. 573-598 | DOI | MR | Zbl

[Cay69] A. Cayley - “VII. A memoir on cubic surfaces”, Philos. Trans. Roy. Soc. London 159 (1869), p. 231-326 | DOI

[CB03] W. Crawley-Boevey - “On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero”, Duke Math. J. 118 (2003) no. 2, p. 339-352 | DOI | MR | Zbl

[CB04] W. Crawley-Boevey - “Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity”, Publ. Math. Inst. Hautes Études Sci. (2004) no. 100, p. 171-207 | DOI | MR | Zbl

[CBS06] W. Crawley-Boevey & P. Shaw - “Multiplicative preprojective algebras, middle convolution and the Deligne–Simpson problem”, Adv. Math. 201 (2006) no. 1, p. 180-208 | DOI | MR | Zbl

[CDDP15] W.-y. Chuang, D.-E. Diaconescu, R. Donagi & T. Pantev - “Parabolic refined invariants and Macdonald polynomials”, Comm. Math. Phys. 335 (2015) no. 3, p. 1323-1379 | DOI | MR | Zbl

[CDP14] W. Y. Chuang, D. E. Diaconescu & G. Pan - “BPS states and the P=W conjecture”, in Moduli spaces, London Math. Soc. Lecture Note Ser., vol. 411, Cambridge Univ. Press, Cambridge, 2014, p. 132-150 | DOI | MR | Zbl

[CL09] S. Cantat & F. Loray - “Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation”, Ann. Inst. Fourier (Grenoble) 59 (2009) no. 7, p. 2927-2978 | DOI | Numdam | Zbl

[CL16] P.-H. Chaudouard & G. Laumon - “Sur le comptage des fibrés de Hitchin nilpotents”, J. Inst. Math. Jussieu 15 (2016) no. 1, p. 91–164 | DOI | Zbl

[Cor88] K. Corlette - “Flat $G$ -bundles with canonical metrics”, J. Differential Geom. 28 (1988) no. 3, p. 361-382 | DOI | MR | Zbl

[CRV18] E. Carlsson & F. Rodriguez Villegas - “Vertex operators and character varieties”, Adv. Math. 330 (2018), p. 38-60 | DOI | MR | Zbl

[dCHM12] M. de Cataldo, T. Hausel & L. Migliorini - “Topology of Hitchin systems and Hodge theory of character varieties: the case $A_{1}$ ”, Ann. of Math. (2) 175 (2012) no. 3, p. 1329-1407 | DOI | MR | Zbl

[dCMS22] M. de Cataldo, D. Maulik & J. Shen - “Hitchin fibrations, abelian surfaces, and the P=W conjecture”, J. Amer. Math. Soc. 35 (2022) no. 3, p. 911-953 | DOI | MR | Zbl

[DDP18] D.-E. Diaconescu, R. Donagi & T. Pantev - “BPS states, torus links and wild character varieties”, Comm. Math. Phys. 359 (2018) no. 3, p. 1027-1078 | DOI | MR | Zbl

[Del70] P. Deligne - Équations différentielles à points singuliers réguliers, Lect. Notes in Math., vol. 163, Springer-Verlag, Berlin, Heidelberg, New York, 1970 | DOI

[Del71] P. Deligne - “Théorie de Hodge: II”, Publ. Math. Inst. Hautes Études Sci. 40 (1971), p. 5-57 | DOI | Numdam | Zbl

[Dia18] D.-E. Diaconescu - “Local curves, wild character varieties, and degenerations”, Commun. Number Theory Phys. 12 (2018) no. 3, p. 491-542 | DOI | MR | Zbl

[Don87] S. K. Donaldson - “Twisted harmonic maps and the self-duality equations”, Proc. London Math. Soc. (3) 55 (1987) no. 1, p. 127-131 | DOI | MR | Zbl

[EOR04] P. Etingof, A. Oblomkov & E. Rains - “Generalized double affine Hecke algebras of rank $1$ and quantized Del Pezzo surfaces”, Adv. Math. 212 (2004) no. 2, p. 749-796 | DOI | MR | Zbl

[Fel21] C. Felisetti - “Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve”, J. Inst. Math. Jussieu (2021), 50 p., online | DOI

[FK97] Fricke & Klein - Vorlesungen über die Theorie der automorphen Funktionen, B. G. Teubner, Leipzig, 1897

[FM22] C. Felisetti & M. Mauri - “P=W conjectures for character varieties with symplectic resolution”, J. Éc. polytech. Math. 9 (2022), p. 853-905 | DOI | MR | Zbl

[FSS18] R. Fedorov, A. Soibelman & Y. Soibelman - “Motivic classes of moduli of Higgs bundles and moduli of bundles with connections”, Commun. Number Theory Phys. 12 (2018) no. 4, p. 687-766 | DOI | MR | Zbl

[FSS20] R. Fedorov, A. Soibelman & Y. Soibelman - “Motivic Donaldson-Thomas invariants of parabolic Higgs bundles and parabolic connections on a curve”, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), article ID 070, 49 pages | DOI | MR | Zbl

[GM83] M. Goresky & R. MacPherson - “Intersection homology. II”, Invent. Math. 72 (1983) no. 1, p. 77-129 | DOI | MR | Zbl

[GO19] P. Gothen & A. G. Oliveira - “Topological mirror symmetry for parabolic Higgs bundles”, J. Geom. Phys. 137 (2019), p. 7-34 | DOI | MR | Zbl

[Gol09] W. M. Goldman - “Trace coordinates on Fricke spaces of some simple hyperbolic surfaces”, in Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, European Mathematical Society, Zürich, 2009, p. 611-684 | DOI | MR | Zbl

[Got94] P. Gothen - “The Betti numbers of the moduli space of rank $3$ Higgs bundles on a Riemann surface”, Internat. J. Math. 5 (1994), p. 861-875 | DOI | MR | Zbl

[GP18] Á. González-Prieto - Topological quantum field theories for character varieties, Ph. D. Thesis, Universidad Complutense de Madrid, 2018

[GPGM07] O. García-Prada, P. Gothen & V. Muñoz - Betti numbers of the moduli space of rank $3$ parabolic Higgs bundles, Mem. Amer. Math. Soc., vol. 187, no. 879, American Mathematical Society, Providence, RI, 2007 | DOI

[GPH13] O. García-Prada & J. Heinloth - “The $y$ -genus of the moduli space of ${PGL}_{n}$ -Higgs bundles on a curve (for degree coprime to $n$ )”, Duke Math. J. 162 (2013) no. 14, p. 2731-2749 | DOI | MR | Zbl

[GPHS14] O. García-Prada, J. Heinloth & A. Schmitt - “On the motives of moduli of chains and Higgs bundles”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 12, p. 2617-2668 | DOI | MR | Zbl

[GWZ20] M. Groechenig, D. Wyss & P. Ziegler - “Mirror symmetry for moduli spaces of Higgs bundles via $p$ -adic integration”, Invent. Math. 221 (2020) no. 2, p. 505-596 | DOI | MR | Zbl

[Har77] R. Hartshorne - Algebraic Geometry, Graduate Texts in Math., Springer, New York, 1977 | DOI

[Hau98] T. Hausel - “Compactification of moduli of Higgs bundles”, J. reine angew. Math. 503 (1998), p. 169-192 | DOI | MR | Zbl

[Hau05] T. Hausel - “Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve”, in Geometric methods in algebra and number theory, Progress in Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, p. 193-217 | DOI | MR | Zbl

[Hit87] N. J. Hitchin - “The self-duality equations on a Riemann surface”, Proc. London Math. Soc. (3) 55 (1987) no. 1, p. 59-126 | DOI | MR | Zbl

[HKLR87] N. J. Hitchin, A. Karlhede, U. Lindström & M. Roček - “HyperKähler metrics and supersymmetry”, Comm. Math. Phys. 108 (1987), p. 535-589 | DOI | Zbl

[HLRV11] T. Hausel, E. Letellier & F. Rodriguez Villegas - “Arithmetic harmonic analysis on character and quiver varieties”, Duke Math. J. 160 (2011) no. 2, p. 323-400 | DOI | MR | Zbl

[HRV08] T. Hausel & F. Rodriguez Villegas - “Mixed Hodge polynomials of character varieties”, Invent. Math. 174 (2008) no. 3, p. 555-624 | DOI | MR | Zbl

[HT01] T. Hausel & M. Thaddeus - “Examples of mirror partners arising from integrable systems”, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) no. 4, p. 313-318 | DOI | MR | Zbl

[HT03a] T. Hausel & M. Thaddeus - “Mirror symmetry, Langlands duality, and the Hitchin system”, Invent. Math. 153 (2003) no. 1, p. 197-229 | DOI | MR | Zbl

[HT03b] T. Hausel & M. Thaddeus - “Relations in the cohomology ring of the moduli space of rank $2$ Higgs bundles”, J. Amer. Math. Soc. 16 (2003) no. 2, p. 303-329 | DOI | MR | Zbl

[HT04] T. Hausel & M. Thaddeus - “Generators for the cohomology ring of the moduli space of rank $2$ Higgs bundles”, Proc. London Math. Soc. (3) 88 (2004) no. 3, p. 632–658 | DOI | MR | Zbl

[Hun96] B. Hunt - The geometry of some special arithmetic quotients, Lect. Notes in Math., vol. 1637, Springer-Verlag, Berlin, 1996 | DOI

[IIS06a] M.-a. Inaba, K. Iwasaki & M.-H. Saito - “Dynamics of the sixth Painlevé equation”, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., vol. 14, Société Mathématique de France, Paris, 2006, p. 103-167 | Zbl

[IIS06b] M.-a. Inaba, K. Iwasaki & M.-H. Saito - “Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part I”, Publ. RIMS, Kyoto Univ. 42 (2006) no. 4, p. 987-1089 | DOI | Zbl

[IIS06c] M.-a. Inaba, K. Iwasaki & M.-H. Saito - “Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II”, in Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, p. 387-432 | DOI | Zbl

[Ina13] M.-a. Inaba - “Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence”, J. Algebraic Geom. 22 (2013) no. 3, p. 407-480 | DOI | MR | Zbl

[Kat96] N. M. Katz - Rigid local systems, Ann. of Math. studies, vol. 139, Princeton University Press, Princeton, NJ, 1996 | DOI | MR

[Kon93] H. Konno - “Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface”, J. Math. Soc. Japan 45 (1993) no. 2, p. 253-276 | DOI | MR | Zbl

[Kos04] V. P. Kostov - “The Deligne–Simpson problem—a survey”, J. Algebra 281 (2004) no. 1, p. 83-108 | DOI | MR | Zbl

[KP81] H. Kraft & C. Procesi - “Minimal singularities in ${GL}_{n}$ ”, Invent. Math. 62 (1981) no. 3, p. 503-515 | DOI | MR | Zbl

[Kur65] M. Kuranishi - “New proof for the existence of locally complete families of complex structures”, in Proceedings of the conference on complex analysis (A. Aeppli, E. Calabi & H. Röhrl, eds.), Springer, Berlin, Heidelberg, 1965, p. 142-154 | DOI

[Let13] E. Letellier - “Quiver varieties and the character ring of general linear groups over finite fields”, J. Eur. Math. Soc. (JEMS) 15 (2013) no. 4, p. 1375-1455 | DOI | MR | Zbl

[Let15] E. Letellier - “Character varieties with Zariski closures of ${GL}_{n}$ conjugacy classes at punctures”, Selecta Math. (N.S.) 21 (2015) no. 1, p. 293-344 | DOI | MR | Zbl

[LM10] M. Logares & J. Martens - “Moduli of parabolic Higgs bundles and Atiyah algebroids”, J. reine angew. Math. 649 (2010), p. 89-116 | DOI | MR | Zbl

[LM14] M. Logares & V. Muñoz - “Hodge polynomials of the $SL (2, ℂ)$ -character variety of an elliptic curve with two marked points”, Internat. J. Math. 25 (2014) no. 14, p. 1450125, 22 | DOI | MR | Zbl

[LMN13] M. Logares, V. Muñoz & P. E. Newstead - “Hodge polynomials of $SL (2, ℂ)$ -character varieties for curves of small genus”, Rev. Mat. Univ. Complut. Madrid 26 (2013) no. 2, p. 635-703 | DOI | MR | Zbl

[Lus81] G. Lusztig - “Green polynomials and singularities of unipotent classes”, Adv. Math. 42 (1981) no. 2, p. 169-178 | DOI | MR | Zbl

[Lus84] G. Lusztig - “Intersection cohomology complexes on a reductive group”, Invent. Math. 75 (1984) no. 2, p. 205-272 | DOI | MR | Zbl

[Lus85] G. Lusztig - “Character sheaves I”, Adv. Math. 56 (1985) no. 3, p. 193-237 | DOI | MR | Zbl

[Lus86] G. Lusztig - “On the character values of finite Chevalley groups at unipotent elements”, J. Algebra 104 (1986) no. 1, p. 146-194 | DOI | MR | Zbl

[LW21] F. Loeser & D. Wyss - “Motivic integration on the Hitchin fibration”, Algebraic Geom. 8 (2021) no. 2, p. 196-230 | DOI | MR | Zbl

[Mac15] I. G. Macdonald - Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015

[Mar17] J. Martínez - “E-polynomials of $PGL (2, ℂ)$ -character varieties of surface groups”, 2017 | arXiv

[Mau21a] M. Mauri - “Intersection cohomology of rank $2$ character varieties of surface groups”, J. Inst. Math. Jussieu (2021), 40 p., online | DOI

[Mau21b] M. Mauri - “Topological mirror symmetry for rank two character varieties of surface groups”, Abh. Math. Sem. Univ. Hamburg 91 (2021) no. 2, p. 297-303 | DOI | MR | Zbl

[Mel19] A. Mellit - “Cell decompositions of character varieties”, 2019 | arXiv

[Mel20a] A. Mellit - “Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers”, Ann. of Math. (2) 192 (2020) no. 1, p. 165-228 | DOI | Zbl

[Mel20b] A. Mellit - “Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)”, Invent. Math. 221 (2020) no. 1, p. 301-327 | DOI | Zbl

[Mer15] M. Mereb - “On the E-polynomials of a family of character varieties”, Math. Ann. 363 (2015) no. 3-4, p. 857-892 | DOI | MR | Zbl

[MH74] Y. I. Manin & M. Hazewinkel - Cubic forms: algebra, geometry, arithmetic, North-Holland Math. Library, vol. 4, American Elsevier Publishing Co., Inc., New York, 1974

[MM16a] J. Martínez & V. Muñoz - “E-polynomials of $SL (2, ℂ)$ -character varieties of complex curves of genus 3” 53, 2016 no. 3, p. 645-681 | MR | Zbl

[MM16b] J. Martínez & V. Muñoz - “E-polynomials of the $SL (2, ℂ)$ -character varieties of surface groups”, Internat. Math. Res. Notices (2016) no. 3, p. 926-961 | DOI | MR | Zbl

[Moz12] S. Mozgovoy - “Solutions of the motivic ADHM recursion formula”, Internat. Math. Res. Notices (2012) no. 18, p. 4218-4244 | DOI | MR | Zbl

[MS80] V. Mehta & C. Seshadri - “Moduli of vector bundles on curves with parabolic structures”, Math. Ann. 248 (1980) no. 3, p. 205-239 | DOI | MR | Zbl

[MS14] S. Mozgovoy & O. Schiffmann - “Counting Higgs bundles”, 2014 | arXiv

[MS20] S. Mozgovoy & O. Schiffmann - “Counting Higgs bundles and type $A$ quiver bundles”, Compositio Math. 156 (2020) no. 4, p. 744–769 | DOI | MR | Zbl

[Nak96] H. Nakajima - “Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces”, in Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Appl. Math., vol. 179, Dekker, New York, 1996, p. 199-208 | Zbl

[Nak98] H. Nakajima - “Quiver varieties and Kac-Moody algebras”, Duke Math. J. 91 (1998) no. 3, p. 515-560 | DOI | MR | Zbl

[Nak01] H. Nakajima - “Quiver varieties and finite dimensional representations of quantum affine algebras”, J. Amer. Math. Soc. 14 (2001) no. 1, p. 145-238 | DOI | MR | Zbl

[NS65] M. S. Narasimhan & C. S. Seshadri - “Stable and unitary vector bundles on a compact Riemann surface”, Ann. of Math. (2) 82 (1965) no. 3, p. 540-567 | DOI | MR | Zbl

[Pau96] C. Pauly - “Espaces de modules de fibrés paraboliques et blocs conformes”, Duke Math. J. 84 (1996) no. 1, p. 217-235 | DOI | Zbl

[Sai86] M. Saito - “Mixed Hodge modules”, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986) no. 9, p. 360-363 | DOI | MR | Zbl

[Sch16] O. Schiffmann - “Indecomposable vector bundles and stable Higgs bundles over smooth projective curves”, Ann. of Math. (2) 183 (2016) no. 1, p. 297-362 | DOI | MR | Zbl

[Shm12] D. A. Shmelkin - “Some remarks on Nakajima’s quiver varieties of type $A$ ”, in Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Société Mathématique de France, Paris, 2012, p. 419-427 | MR | Zbl

[Sim88] C. Simpson - “Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization”, J. Amer. Math. Soc. 1 (1988) no. 4, p. 867-918 | DOI | MR | Zbl

[Sim90] C. Simpson - “Harmonic bundles on noncompact curves”, J. Amer. Math. Soc. 3 (1990) no. 3, p. 713-770 | DOI | MR | Zbl

[Sim92] C. Simpson - “Higgs bundles and local systems”, Publ. Math. Inst. Hautes Études Sci. 75 (1992), p. 5-95 | DOI | Numdam | Zbl

[Sim94a] C. Simpson - “Moduli of representations of the fundamental group of a smooth projective variety I”, Publ. Math. Inst. Hautes Études Sci. 79 (1994), p. 47-129 | DOI | Numdam | Zbl

[Sim94b] C. Simpson - “Moduli of representations of the fundamental group of a smooth projective variety II”, Publ. Math. Inst. Hautes Études Sci. 80 (1994), p. 5-79 | DOI | Numdam

[Sim09] C. Simpson - “Katz’s middle convolution algorithm”, Pure Appl. Math. Q 5 (2009), p. 781-852 | DOI | MR | Zbl

[Sim16] C. Simpson - “The dual boundary complex of the ${SL}_{2}$ character variety of a punctured sphere”, Ann. Fac. Sci. Toulouse Math. (6) 25 (2016) no. 2-3, p. 317-361 | DOI | Zbl

[Soi16] A. Soibelman - “Parabolic bundles over the projective line and the Deligne–Simpson problems”, Q. J. Math. 67 (2016) no. 1, p. 75-108 | DOI | MR | Zbl

[Soi18] A. Soibelman - “The very good property for parabolic vector bundles over curves”, Lett. Math. Phys. 108 (2018) no. 6, p. 1551-1561 | DOI | MR | Zbl

[Spr76] T. Springer - “Trigonometric sums, Green functions of finite groups and representations of Weyl groups”, Invent. Math. 36 (1976), p. 173-207 | DOI | MR | Zbl

[ST22] T. Schedler & A. Tirelli - “Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces”, in Representation theory and algebraic geometry, Springer, Cham, 2022, p. 393-459 | DOI | Zbl

[Vog89] H. Vogt - “Sur les invariants fondamentaux des équations différentielles linéaires du second ordre”, Ann. Sci. École Norm. Sup. 6 (1889), p. 3-71 | DOI | Numdam | Zbl

[Yam08] D. Yamakawa - “Geometry of multiplicative preprojective algebra”, Internat. Math. Res. Papers (2008), article ID rpn008, 77 pages | DOI | MR | Zbl

[Yok93] K. Yokogawa - “Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves”, J. Math. Kyoto Univ. 33 (1993) no. 2, p. 451-504 | DOI | MR | Zbl

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