Abstract
Let p be a prime number. We prove that the \(P=W\) conjecture for \(\mathrm {SL}_p\) is equivalent to the \(P=W\) conjecture for \(\mathrm {GL}_p\). As a consequence, we verify the \(P=W\) conjecture for genus 2 and \(\mathrm {SL}_p\). For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the \(\mathrm {SL}_p\)-Hitchin moduli space and the \(\mathrm {SL}_p\)-twisted character variety, relying on Gröchenig–Wyss–Ziegler’s recent proof of the topological mirror conjecture by Hausel–Thaddeus. Finally we discuss obstructions of studying the cohomology of the \(\mathrm {SL}_n\)-Hitchin moduli space via compact hyper-Kähler manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We call M an irreducible hyper-Kähler manifold if M is simply connected satisfying that \(H^0(M, \Omega _M)\) is generated by a non-where degenerate holomorphic 2-form. We say that a hyper-Kähler manifold is of Kummer type if it deforms to a generalized Kummer variety.
Since this construction is not essentially used in the present paper, we omit further details.
References
Beauville, A.: Variétés kählériennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1984)
Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100. Soc. Math. France, Paris (1982)
de Cataldo, M.A., Hausel, T., Migliorini, L.: Topology of Hitchin systems and Hodge theory of character varieties: the case \(A_1\). Ann. Math. (2) 175(3), 1329–1407 (2012)
de Cataldo, M.A., Maulik, D., Shen, J.: Hitchin fibrations, abelian surfaces, and the \(P=W\) conjecture. J. Am. Math. Soc. 35(3), 911–953 (2022)
de Cataldo, M.A., Migliorini, L.: The perverse filtration and the Lefschetz hyperplane section theorem. Ann. Math. 171(3), 2089–2113 (2010)
Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286, 193–207 (1990)
Green, M., Kim, Y.-J., Laza, R., Robles, C.: The LLV decomposition of hyper–Kähler cohomology (the known cases and the general conjectural behavior). arXiv:1906.03432v2. Math. Ann., to appear
Groechenig, M., Wyss, D., Ziegler, P.: Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration. Invent. Math. 221(2), 505–596 (2022)
Groechenig, M., Wyss, D., Ziegler, P.: Geometric stabilisation via \(p\)-adic integration. J. Amer. Math. Soc. 33, 807–873 (2020)
Donagi, R., Ein, L., Lazarsfeld, R.: Nilpotent cones and sheaves on K3 surfaces, Birational algebraic geometry (Baltimore, MD, 1996), pp. 51–61. Contemp. Math., 207, Amer. Math. Soc., Providence (1997)
Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. 160(2), 323–400 (2011)
Hausel, T., Pauly, C.: Prym varieties of spectral covers. Geom. Topol. 16(3), 1609–1638 (2012)
Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties, with an appendix by Nicholas M. Katz. Invent. Math. 174(3), 555–624 (2008)
Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153(1), 197–229 (2003)
Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987)
Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54, 91–114 (1987)
Looijenga, E., Lunts, V.A.: A Lie algebra attached to a projective variety. Invent. Math. 129(2), 361–412 (1997)
Markman, E.: Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. J. Reine Angew. Math. 544, 61–82 (2002)
Maulik, D., Shen, J.: Endoscopic decompositions and the Hausel–Thaddeus conjecture. Forum Math. Pi, 9, No. e8 (2021)
Mereb, M.: On the E -polynomials of a family of \(\rm Sl_n\)-character varieties. Math. Ann. 363(3–4), 857–892 (2015)
Mongardi, G., Rapagnetta, A., Saccá, G.: The Hodge diamond of O’Grady’s six-dimensional example. Compos. Math. 154(5), 984–1013 (2018)
O’Grady, K.G.: Desingularized moduli spaces of sheaves on a K3. J. Reine Angew. Math. 512, 49–117 (1999)
O’Grady, K.G.: A new six-dimensional irreducible symplectic variety. J. Algebraic Geom. 12(3), 435–505 (2003)
Shende, V.: The weights of the tautological classes of character varieties. Int. Math. Res. Not. 22, 6832–6840 (2017)
Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. No. 75, 5–95 (1992)
Simpson, C.T.: Moduli of representations of the fundamental group of smooth projective varieties II," Inst. Hautes. Etudes Sci. Publ. Math. No. 80(1994), 5–79 (1995)
Verbitsky, M.: Cohomology of compact hyperkaehler manifolds. Thesis (Ph.D.)–Harvard University (1995)
Verbitsky, M.: Cohomology of compact hyper-Kähler manifolds and its applications. Geom. Funct. Anal. 6(4), 601–611 (1996)
Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)
Acknowledgements
We are grateful to Chen Wan and Zhiwei Yun for helpful discussions. We also thank the anonymous referee for careful reading of the paper. The first-named author is partially supported by NSF DMS Grant 1901975. The third-named author is partially supported by NSF DMS Grant 2134315.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
