P=W conjectures for character varieties with symplectic resolution
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 853-905.

We establish P=W and PI=WI conjectures for character varieties with structural group GL n and SL n which admit a symplectic resolution, i.e., for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for a resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is a specialization of the singular irreducible holomorphic symplectic variety of type O’Grady 6.

On établit les conjectures P=W et PI=WI pour les variétés de caractères avec groupe structurel GL n et SL n qui admettent une résolution symplectique, c’est-à-dire pour le genre 1 en rang arbitraire, et le genre 2 en rang 2. On formule la conjecture P=W pour une résolution et on la prouve pour les résolutions symplectiques. Pour la démonstration on fait appel à la topologie des modifications birationnelles et quasi-étales des espaces de modules de fibrés de Higgs. Pour cela, on démontre des résultats auxiliaires d’intérêt indépendant, comme la construction d’une compactification relative de l’espace de modules de Hodge pour les groupes algébriques réductifs, ou la théorie de l’intersection de certains cycles lagrangiens singuliers. En particulier, on étudie en détail un espace de modules des fibrés de Higgs qui est une spécialisation de la variété symplectique holomorphe irréductible singulière de type O’Grady 6.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.196
Classification: 14D20, 53D30, 14D22, 14E15, 32S35, 55N33
Keywords: P=W conjecture, intersection cohomology, Higgs bundles
Mot clés : Conjecture P=W, cohomologie d’intersection, fibrés de Higgs
Camilla Felisetti 1; Mirko Mauri 2

1 University of Trento, Department of mathematics Via Sommarive 14, 38123 Povo (TN), Italy
2 University of Michigan East Hall, 530 Church St, Ann Arbor 48109, Michigan, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2022__9__853_0,
     author = {Camilla Felisetti and Mirko Mauri},
     title = {P=W conjectures for character varieties with symplectic resolution},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {853--905},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.196},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.196/}
}
TY  - JOUR
AU  - Camilla Felisetti
AU  - Mirko Mauri
TI  - P=W conjectures for character varieties with symplectic resolution
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2022
SP  - 853
EP  - 905
VL  - 9
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.196/
DO  - 10.5802/jep.196
LA  - en
ID  - JEP_2022__9__853_0
ER  - 
%0 Journal Article
%A Camilla Felisetti
%A Mirko Mauri
%T P=W conjectures for character varieties with symplectic resolution
%J Journal de l’École polytechnique — Mathématiques
%D 2022
%P 853-905
%V 9
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.196/
%R 10.5802/jep.196
%G en
%F JEP_2022__9__853_0
Camilla Felisetti; Mirko Mauri. P=W conjectures for character varieties with symplectic resolution. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 853-905. doi : 10.5802/jep.196. https://jep.centre-mersenne.org/articles/10.5802/jep.196/

[1] E. Amerik & M. Verbitsky - “Contraction centers in families of hyperkähler manifolds”, Selecta Math. (N.S.) 27 (2021) no. 4, article ID 60, 26 pages | DOI | Zbl

[2] A. Beauville - “Variétés kähleriennes dont la première classe de Chern est nulle”, J. Differential Geom. 18 (1983) no. 4, p. 755-782 (1984) | Zbl

[3] A. Beauville, M. S. Narasimhan & S. Ramanan - “Spectral curves and the generalised theta divisor”, J. reine angew. Math. 398 (1989), p. 169-179 | DOI | MR | Zbl

[4] A. A. Beilinson, J. N. Bernstein & P. Deligne - “Faisceaux pervers”, in Analysis and topology on singular spaces, I (Luminy,1981), Astérisque, vol. 100, Société Mathématique de France, Paris, 1982, p. 5-171 | MR | Zbl

[5] G. Bellamy & T. Schedler - “Symplectic resolutions of character varieties”, 2019 | arXiv

[6] A. Białynicki-Birula - “Some theorems on actions of algebraic groups”, Ann. of Math. (2) 98 (1973), p. 480-497 | DOI | MR | Zbl

[7] E. Bierstone, P. D. Milman & M. Temkin - “-universal desingularization”, Asian J. Math. 15 (2011) no. 2, p. 229-249 | DOI | MR

[8] I. Biswas - “Orbifold principal bundles on an elliptic fibration and parabolic principal bundles on a Riemann surface. II”, Collect. Math. 56 (2005) no. 3, p. 235-252 | MR | Zbl

[9] I. Biswas, S. Majumder & M. L. Wong - “Parabolic Higgs bundles and Γ-Higgs bundles”, J. Austral. Math. Soc. 95 (2013) no. 3, p. 315-328 | DOI | MR | Zbl

[10] M. A. de Cataldo - “Projective compactification of Dolbeault moduli spaces”, Internat. Math. Res. Notices (2021) no. 5, p. 3543-3570 | DOI | MR | Zbl

[11] M. A. 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

[12] M. A. de Cataldo, T. Hausel & L. Migliorini - “Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces”, J. Singul. 7 (2013), p. 23-38 | DOI | MR | Zbl

[13] M. A. de Cataldo & D. Maulik - “The perverse filtration for the Hitchin fibration is locally constant”, Pure Appl. Math. Q 16 (2020) no. 5, p. 1441-1464 | DOI | MR

[14] M. A. de Cataldo, D. Maulik & J. Shen - “On the P=W conjecture for SL n , 2020 | arXiv

[15] M. A. de Cataldo, D. Maulik & J. Shen - “Hitchin fibrations, abelian surfaces, and the P=W conjecture”, J. Amer. Math. Soc. (2021), online, 47 p. | DOI

[16] M. A. de Cataldo & L. Migliorini - “The Hodge theory of algebraic maps”, Ann. Sci. École Norm. Sup. (4) 38 (2005) no. 5, p. 693-750 | DOI | Numdam | MR | Zbl

[17] M. A. de Cataldo & L. Migliorini - “The perverse filtration and the Lefschetz hyperplane theorem”, Ann. of Math. (2) 171 (2010) no. 3, p. 2089-2113 | DOI | MR | Zbl

[18] I. Cheltsov, V. Przyjalkowski & C. Shramov - “Which quartic double solids are rational?”, J. Algebraic Geom. 28 (2019) no. 2, p. 201-243 | DOI | MR | Zbl

[19] S. M. Chiarello, T. Hausel & A. Szenes - “An enumerative approach to P=W, 2020 | arXiv

[20] G. D. Daskalopoulos & K. K. Uhlenbeck - “An application of transversality to the topology of the moduli space of stable bundles”, Topology 34 (1995) no. 1, p. 203-215 | DOI | MR | Zbl

[21] G. D. Daskalopoulos, R. A. Wentworth & G. Wilkin - “Cohomology of SL (2,) character varieties of surface groups and the action of the Torelli group”, Asian J. Math. 14 (2010) no. 3, p. 359-383 | DOI | MR | Zbl

[22] P. Deligne - “Théorie de Hodge. III”, Publ. Math. Inst. Hautes Études Sci. (1974) no. 44, p. 5-77 | DOI | Numdam | Zbl

[23] R. Donagi, L. Ein & R. Lazarsfeld - “Nilpotent cones and sheaves on K3 surfaces”, in Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math., vol. 207, American Mathematical Society, Providence, RI, 1997, p. 51-61 | DOI | MR | Zbl

[24] J.-M. Drezet & M. S. Narasimhan - “Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques”, Invent. Math. 97 (1989) no. 1, p. 53-94 | DOI | Zbl

[25] A. H. Durfee - “Intersection homology Betti numbers”, Proc. Amer. Math. Soc. 123 (1995) no. 4, p. 989-993 | DOI | MR | Zbl

[26] C. Felisetti - “Intersection cohomology of the moduli space of Higgs bundles on a genus 2 curve”, J. Inst. Math. Jussieu (2021), p. 1-50 | DOI

[27] C. Felisetti, J. Shen & Q. Yin - “On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties”, Trans. Amer. Math. Soc. 375 (2022) no. 4, p. 2987-3001 | DOI | MR | Zbl

[28] E. Franco, O. Garcia-Prada & P. E. Newstead - “Higgs bundles over elliptic curves”, Illinois J. Math. 58 (2014) no. 1, p. 43-96 | MR | Zbl

[29] B. Fu & Y. Namikawa - “Uniqueness of crepant resolutions and symplectic singularities”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 1, p. 1-19 | DOI | MR | Zbl

[30] M. Goresky & R. MacPherson - “Intersection homology theory”, Topology 19 (1980) no. 2, p. 135-162 | DOI | MR | Zbl

[31] M. Goresky & R. MacPherson - “Morse theory and intersection homology theory”, in Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, vol. 101, Société Mathématique de France, Paris, 1983, p. 135-192 | MR | Zbl

[32] L. Göttsche & W. Soergel - “Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces”, Math. Ann. 296 (1993) no. 2, p. 235-245 | DOI | MR | Zbl

[33] P. Graf & M. Schwald - “On the Kodaira problem for uniruled Kähler spaces”, Ark. Mat. 58 (2020) no. 2, p. 267-284 | DOI | Zbl

[34] D. Greb, S. Kebekus & S. Kovács - “Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties”, Compositio Math. 146 (2010) no. 1, p. 193-219 | DOI | MR | Zbl

[35] M. Groechenig - “Hilbert schemes as moduli of Higgs bundles and local systems”, Internat. Math. Res. Notices (2014) no. 23, p. 6523-6575 | DOI | MR | Zbl

[36] A. Grothendieck - “Sur quelques points d’algèbre homologique”, Tôhoku Math. J. 9 (1957), p. 119-221 | DOI | Zbl

[37] A. Harder - “Torus fibers and the weight filtration”, 2019 | arXiv

[38] A. Harder, L. Zhiyuan, J. Shen & Q. Yin - “P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds”, Forum Math. Sigma 9 (2021), article ID e50 | DOI | Zbl

[39] T. Hausel - “Vanishing of intersection numbers on the moduli space of Higgs bundles”, Adv. Theo. Math. Phys. 2 (1998) no. 5, p. 1011-1040 | DOI | MR | Zbl

[40] 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

[41] T. Hausel & F. Rodriguez-Villegas - “Mixed Hodge polynomials of character varieties”, Invent. Math. 174 (2008) no. 3, p. 555-624, With an appendix by Nicholas M. Katz | DOI | MR | Zbl

[42] T. Hausel & F. Rodriguez-Villegas - “Cohomology of large semiprojective hyperkähler varieties”, in De la géométrie algébrique aux formes automorphes, Astérisque, vol. 370, Société Mathématique de France, Paris, 2015, p. 113-156 | Zbl

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

[44] 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

[45] V. Heu & F. Loray - Flat rank two vector bundles on genus two curves, Mem. Amer. Math. Soc., vol. 259, no. 1247, American Mathematical Society, Providence, RI, 2019

[46] 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

[47] D. Kaledin - “Symplectic singularities from the Poisson point of view”, J. reine angew. Math. 600 (2006), p. 135-156 | DOI | MR | Zbl

[48] L. Katzarkov, V. V. Przyjalkowski & A. Harder - “P=W phenomena”, Mat. Zametki 108 (2020) no. 1, p. 33-46 | DOI | Zbl

[49] Y.-H. Kiem & S.-B. Yoo - “The stringy E-function of the moduli space of Higgs bundles with trivial determinant”, Math. Nachr. 281 (2008) no. 6, p. 817-838 | DOI | MR | Zbl

[50] F. Kirwan & J. Woolf - An introduction to intersection homology theory, Chapman & Hall/CRC, Boca Raton, FL, 2006 | DOI

[51] J. Kollár - Lectures on resolution of singularities, Annals of Math. Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007

[52] J. Kollár - Singularities of the minimal model program, Cambridge Tracts in Math., vol. 200, Cambridge University Press, Cambridge, 2013 | DOI

[53] J. Kollár & S. Mori - Birational geometry of algebraic varieties, Cambridge Tracts in Math., vol. 134, Cambridge University Press, Cambridge, 1998 | DOI

[54] C. Kumar - “Invariant vector bundles of rank 2 on hyperelliptic curves”, Michigan Math. J. 47 (2000) no. 3, p. 575-584 | DOI | MR | Zbl

[55] R. Lazarsfeld - Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer-Verlag, Berlin, 2004 | DOI

[56] M. Lehn & C. Sorger - “La singularité de O’Grady”, J. Algebraic Geom. 15 (2006) no. 4, p. 753-770 | DOI | Zbl

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

[58] D. Maulik & A. Okounkov - Quantum groups and quantum cohomology, Astérisque, vol. 408, Société Mathématique de France, Paris, 2019 | DOI

[59] M. Mauri - “Intersection cohomology of rank 2 character varieties of surface groups”, J. Inst. Math. Jussieu (2021), p. 1–40

[60] M. Mauri, E. Mazzon & M. Stevenson - “On the geometric P=W conjecture”, Selecta Math. (N.S.) 28 (2022) no. 3, article ID 65, 45 pages | DOI | MR | Zbl

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

[62] G. Mongardi, A. Rapagnetta & G. Saccà - “The Hodge diamond of O’Grady’s six-dimensional example”, Compositio Math. 154 (2018) no. 5, p. 984-1013 | DOI | MR | Zbl

[63] D. Mumford, J. Fogarty & F. Kirwan - Geometric invariant theory, Ergeb. Math. Grenzgeb. (2), vol. 34, Springer-Verlag, Berlin, 1994 | DOI

[64] M. S. Narasimhan & S. Ramanan - “Moduli of vector bundles on a compact Riemann surface”, Ann. of Math. (2) 89 (1969), p. 14-51 | DOI | MR | Zbl

[65] M. S. Narasimhan & S. Ramanan - “Vector bundles on curves”, in Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, p. 335-346 | Zbl

[66] A. Némethi & S. Szabó - “The geometric P=W conjecture in the Painlevé cases via plumbing calculus”, Internat. Math. Res. Notices (2022) no. 5, p. 3201-3218 | DOI | Zbl

[67] N. Nitsure - “Moduli space of semistable pairs on a curve”, Proc. London Math. Soc. (3) 62 (1991) no. 2, p. 275-300 | DOI | MR | Zbl

[68] S. Pal & C. Pauly - “The wobbly divisors of the moduli space of rank-2 vector bundles”, Adv. Geom. 21 (2021) no. 4, p. 473-482 | DOI | MR | Zbl

[69] A. Perego & A. Rapagnetta - “Deformation of the O’Grady moduli spaces”, J. reine angew. Math. 678 (2013), p. 1-34 | DOI | MR | Zbl

[70] A. Perego & A. Rapagnetta - “The moduli spaces of sheaves on K3 surfaces are irreducible symplectic varieties”, 2018 | arXiv

[71] C. Peters & J. H. M. Steenbrink - Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3), vol. 52, Springer-Verlag, Berlin, 2008

[72] J. Sawon & C. Shen - “Deformations of compact Prym fibrations to Hitchin systems”, 2021 | arXiv

[73] C. S. Seshadri - “Moduli of π-vector bundles over an algebraic curve”, in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, p. 139-260 | MR | Zbl

[74] J. Shen & Z. Zhang - “Perverse filtrations, Hilbert schemes, and the P=W conjecture for parabolic Higgs bundles”, Algebraic Geom. 8 (2021) no. 4, p. 465-489 | DOI | MR | Zbl

[75] J. Shen & Q. Yin - “Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds”, Duke Math. J. 171 (2022) no. 1, p. 209-241 | DOI | Zbl

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

[77] C. T. Simpson - “Higgs bundles and local systems”, Publ. Math. Inst. Hautes Études Sci. (1992) no. 75, p. 5-95 | DOI | MR | Zbl

[78] C. T. Simpson - “Moduli of representations of the fundamental group of a smooth projective variety. I”, Publ. Math. Inst. Hautes Études Sci. (1994) no. 79, p. 47-129 | DOI | Numdam | MR | Zbl

[79] C. T. Simpson - “Moduli of representations of the fundamental group of a smooth projective variety. II”, Publ. Math. Inst. Hautes Études Sci. (1994) no. 80, p. 5-79 (1995) | DOI | Numdam | MR

[80] C. T. Simpson - “The Hodge filtration on nonabelian cohomology”, in Algebraic geometry (Santa Cruz 1995), Proc. Sympos. Pure Math., vol. 62, American Mathematical Society, Providence, RI, 1997, p. 217-281 | DOI | MR | Zbl

[81] S. Szabó - “Simpson’s geometric P=W conjecture in the Painlevé VI case via abelianization”, 2019 | arXiv

[82] S. Szabó - “Perversity equals weight for Painlevé spaces”, Adv. Math. 383 (2021), article ID 107667, 45 pages | DOI | MR | Zbl

[83] M. Temkin - “Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case”, Duke Math. J. 161 (2012) no. 11, p. 2207-2254 | DOI | MR | Zbl

[84] M. Thaddeus - Topology of the moduli space of stable bundles on a Riemann surface, Master’s Thesis, University of Oxford, 1989

[85] A. Weber - “Hirzebruch class and Białynicki-Birula decomposition”, Transform. Groups 22 (2017) no. 2, p. 537-557 | DOI | Zbl

[86] G. Williamson - “Modular representations and reflection subgroups”, in Current developments in math., 2019, International Press, Somerville, MA, 2021, p. 113-184 | DOI | Zbl

[87] B. Wu - “Hodge numbers of O’Grady 6 via Ngô strings”, 2021 | arXiv

[88] K. Yoshioka - “Moduli spaces of stable sheaves on abelian surfaces”, Math. Ann. 321 (2001) no. 4, p. 817-884 | DOI | MR | Zbl

[89] Z. Zhang - “The P=W identity for cluster varieties”, Math. Res. Lett. 28 (2021) no. 3, p. 925-944 | DOI | MR | Zbl

Cited by Sources: