P=W conjectures for character varieties with symplectic resolution

Felisetti, Camilla; Mauri, Mirko

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 853-905.

On établit les conjectures P=W et PI=WI pour les variétés de caractères avec groupe structurel ${\mathrm{GL}}_n$ et ${\mathrm{SL}}_n$ 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 $6$.

We establish P=W and PI=WI conjectures for character varieties with structural group ${\mathrm{GL}}_n$ and ${\mathrm{SL}}_n$ 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 $6$.

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

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

[2] Beauville, A. Variétés kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., Volume 18 (1983) no. 4, p. 755-782 (1984) http://projecteuclid.org/euclid.jdg/1214438181 | Zbl

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

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

[5] Bellamy, G.; Schedler, T. Symplectic resolutions of character varieties, 2019 | arXiv

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

[7] Bierstone, E.; Milman, P. D.; Temkin, M. $\mathbb{Q}$-universal desingularization, Asian J. Math., Volume 15 (2011) no. 2, pp. 229-249 | DOI | MR

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

[9] Biswas, I.; Majumder, S.; Wong, M. L. Parabolic Higgs bundles and $\Gamma$-Higgs bundles, J. Austral. Math. Soc., Volume 95 (2013) no. 3, pp. 315-328 | DOI | MR | Zbl

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

[11] de Cataldo, M. A.; Hausel, T.; Migliorini, L. 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 | Zbl

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

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

[14] de Cataldo, M. A.; Maulik, D.; Shen, J. On the P=W conjecture for ${\mathrm{SL}}_n$, 2020 | arXiv

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

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

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

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

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

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

[21] Daskalopoulos, G. D.; Wentworth, R. A.; Wilkin, G. Cohomology of $\mathrm{SL}(2,ℂ)$ character varieties of surface groups and the action of the Torelli group, Asian J. Math., Volume 14 (2010) no. 3, pp. 359-383 | DOI | MR | Zbl

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

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

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

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

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

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

[28] Franco, E.; Garcia-Prada, O.; Newstead, P. E. Higgs bundles over elliptic curves, Illinois J. Math., Volume 58 (2014) no. 1, pp. 43-96 http://projecteuclid.org/euclid.ijm/1427897168 | MR | Zbl

[29] Fu, B.; Namikawa, Y. Uniqueness of crepant resolutions and symplectic singularities, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 1, pp. 1-19 http://aif.cedram.org/item?id=AIF_2004__54_1_1_0 | DOI | MR | Zbl

[30] Goresky, M.; MacPherson, R. Intersection homology theory, Topology, Volume 19 (1980) no. 2, pp. 135-162 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[41] Hausel, T.; Rodriguez-Villegas, F. 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

[42] Hausel, T.; Rodriguez-Villegas, F. Cohomology of large semiprojective hyperkähler varieties, De la géométrie algébrique aux formes automorphes (Astérisque), Volume 370, Société Mathématique de France, Paris, 2015, pp. 113-156 | Zbl

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

[44] Hausel, T.; Thaddeus, M. Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles, Proc. London Math. Soc. (3), Volume 88 (2004) no. 3, pp. 632-658 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[53] Kollár, J.; Mori, S. Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge University Press, Cambridge, 1998, 254 pages | DOI

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

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

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

[57] Logares, M.; Muñoz, V.; Newstead, P. Hodge polynomials of $\mathrm{SL}(2,ℂ)$-character varieties for curves of small genus, Rev. Mat. Univ. Complut. Madrid, Volume 26 (2013) no. 2, pp. 635-703 | DOI | MR | Zbl

[58] Maulik, D.; Okounkov, A. Quantum groups and quantum cohomology, Astérisque, 408, Société Mathématique de France, Paris, 2019, ix+209 pages | DOI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[75] Shen, Junliang; Yin, Qizheng Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds, Duke Math. J., Volume 171 (2022) no. 1, pp. 209-241 | DOI | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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