Topology of Hitchin systems and Hodge theory of character varieties: the case \(A_1\). (English) Zbl 1375.14047

Let \(\Sigma\) be a non-singular complex projective curve of genus \(g\geq 2\). Let \(\Sigma^{\mathrm{top}}\) be the underlying topological surface. So \(\Sigma\) consists of \(\Sigma^{\mathrm{top}}\) and a complex structure. A Higgs bundle on \(\Sigma\) is a pair \((E,\phi)\) where \(E\) is a holomorphic vector bundle over \(\Sigma\) and \(\phi\in H^0(\Sigma,\mathrm{End}(E)\otimes\Omega)\), called a Higgs field, is a holomorphic endomorphism of \(E\) twisted by the canonical bundle \(\Omega\) of \(\Sigma\). The slope of \(E\) is the degree of \(E\) divided by the rank of \(E\). The bundle is called stable if the slope of any proper \(\phi\)-invariant subbundle is strictly less than that of \(E\). A Higgs bundle is called polystable if it is the direct sum of stable Higgs bundles with the same slope. Fixing the degree \(d\) and the rank \(n\) determines the Dolbeault moduli space of polystable Higgs bundles \(\mathcal{M}_{\mathrm{Dol}}\).
Let \(\Gamma\) be the universal central extension of \(\pi_1(\Sigma)\), and let \(\mathcal{R}:=\mathrm{Hom}^+(\Gamma,\mathrm{GL}(n,\mathbb{C}))\) be the semisimple representations of \(\Gamma\). Then \(\mathrm{GL}(n,\mathbb{C})\) acts on \(\mathcal{R}\) by conjugation and the quotient \(\mathcal{R}/\mathrm{GL}(n,\mathbb{C})\) is called the twisted character variety; it can be realized as a GIT quotient.
Let \(\mathcal{M}_{\mathrm{B}}\) be the subvariety of \(\mathcal{R}/\mathrm{GL}(n,\mathbb{C})\) consisting of characters that map the central extension of \(\Gamma\) to an element in the center of \(\mathrm{SU}(n)\subset \mathrm{GL}(n,\mathbb{C})\), isomorphic to \(\mathbb{Z}_n\), whose least residue corresponds to the degree \(d\). \(\mathcal{M}_{\mathrm{B}}\) is called the Betti moduli space of \(\Sigma\). The non-Abelian Hodge Theorem states that there is an analytic isomorphism between \(\mathcal{M}_{\mathrm{Dol}}\) and \(\mathcal{M}_{\mathrm{B}}\). Since the Betti moduli space does not depend on the complex structure of \(\Sigma\), but the Dolbeault moduli space does, these moduli spaces are not biholomorphic (thus not algebraically isomorphic). Regardless, this diffeomorphism induces an isomorphism on cohomology: \(H^*(\mathcal{M}_{\mathrm{B}})\cong H^*(\mathcal{M}_{\mathrm{Dol}})\).
We note that the non-abelian Hodge Theorem can be generalized to a setting where \(\mathrm{GL}(n,\mathbb{C})\) is replaced by any connected reductive complex algebraic group \(G\). The paper under review restricts its attention to the cases when the degree \(d=1\) and the rank \(n=2\), but also considers the corresponding moduli spaces for \(G=\mathrm{SL}(2,\mathbb{C})\) and \(G=\mathrm{PGL}(2,\mathbb{C})\).
P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 44, 5–77 (1974; Zbl 0237.14003)] showed that a complex variety \(X\) admits an increasing weight filtration \(0=W_{-1}\subset W_0\subset\cdots\subset W_{2j}=H^j(X;\mathbb{Q})\), and a decreasing Hodge filtration \(H^j(X;\mathbb{C})=F^{0}\supset\cdots\supset F^{m+1}=0\) such that for all \(0\leq p\leq l\), \[ \mathrm{Gr}^{W\otimes \mathbb{C}}_l:= W_l\otimes\mathbb{C}/W_{l-1}\otimes\mathbb{C}= F^p(\mathrm{Gr}^{W\otimes \mathbb{C}}_l) \oplus\overline{F^{l-p+1}(\mathrm{Gr}^{W\otimes \mathbb{C}}_l)}, \] where \(F^p(\mathrm{Gr}^{W\otimes \mathbb{C}}_l)=(F^p\cap W_l\otimes \mathbb{C}+W_{l-1}\otimes \mathbb{C})/W_{l-1}\otimes \mathbb{C}\). This allows one to define the mixed Hodge numbers for every \(H^j(X;\mathbb{C})\) and subsequently to define the mixed Hodge polynomial for \(X\).
T. Hausel and F. Rodriguez-Villegas [Invent. Math. 174, No. 3, 555–624 (2008; Zbl 1213.14020)] showed what they call the “curious hard Lefschetz theorem”; namely that \(\mathrm{Gr}^W_{\dim\mathcal{M}_{\mathrm{B}}-2k}H^*(\mathcal{M}_{\mathrm{B}})\cong \mathrm{Gr}^W_{\dim\mathcal{M}_{\mathcal{B}}+2k}H^{*+2k}(\mathcal{M}_{\mathrm{B}})\) for all \(k\geq 0\), where the isomorphism is given by the \(k\)-fold iterated cup product with an explicitly given element in \(H^2(\mathcal{M}_{\mathrm{B}})\). A similar theorem holds when \(G=\mathrm{PGL}(2,\mathbb{C})\). This theorem is proved, in part, by determining the mixed Hodge polynomial of \(\mathcal{M}_{\mathrm{B}}\). This is in contrast to the classical hard Lefschetz theorem since the cohomological degree goes up by \(2k\) and the weight type by \(4k\); hence the usage of “curious”.
In the paper under review, the authors state: “The present paper was partly motivated by the desire to give a more conceptual explanation for these curious hard Lefschetz theorems.”
There is a mapping \(\chi:\mathcal{M}_{\mathrm{Dol}}\to \mathcal{A}\), called the Hitchin fibration, where \(\mathcal{A}\) is non-canonically isomorphic to \(\mathbb{C}^{4g-3}\). Let \(\Lambda^s\subset \mathcal{A}\) be an \(s\)-dimensional linear section with respect to an identification of \(\mathcal{A}\) with \(\mathbb{C}^{4g-3}\); then \[ P_pH^d(\mathcal{M}_{\mathrm{Dol}})=\mathrm{Ker}\{H^d(\mathcal{M}_{\mathrm{Dol}})\to H^d(\chi^{-1}(\Lambda^{d-p-1}))\} \] determines the perverse Leray filtration on \(H^*(\mathcal{M}_{\mathrm{Dol}})\).
The main theorem of this paper establishes that, after renumbering, the weight filtration on \(H^*(\mathcal{M}_{\mathrm{B}})\) exactly corresponds to the perverse Leray filtration on \(H^*(\mathcal{M}_{\mathrm{Dol}})\) under the induced map from the non-abelian Hodge Theorem. In particular, for all \(k\), \[ W_{2k}H^*(\mathcal{M}_{\mathrm{B}})=W_{2k+1}H^*(\mathcal{M}_{\mathrm{B}})=P_kH^*(\mathcal{M}_{\mathrm{Dol}}). \]
Using this correspondence, the authors show that the curious hard Lefschetz theorem on \(H^*(\mathcal{M}_{\mathrm{B}})\), which is in terms of the weight filtration, is explained by a general hard Lefschetz theorem applied to \(H^*(\mathcal{M}_{\mathrm{Dol}})\), which is in terms of the perverse Leray filtration.
This is surprising since this implies, as the authors say, that “the weight filtration on \(H^*(\mathcal{M}_{\mathrm{B}})\) keeps track of certain topological properties of the Hitchin map on \(H^*(\mathcal{M}_{\mathrm{Dol}})\)”. But the weight filtration only depends on \(\Sigma^{\mathrm{top}}\) whereas the Hitchin map depends on \(\Sigma\).
The proof of this deep result is lengthy and technical, but the subject matter appears to demand it. Roughly speaking, by using known results about the cohomology rings, the authors show that the place of the multiplicative generators in the perverse Leray filtration of \(H^*(\mathcal{M}_{\mathrm{Dol}})\) is the same as in the weight filtration of \(H^*(\mathcal{M}_{\mathrm{B}})\). The perverse Leray filtration is generally not compatible with cup products, but the authors show that there is a large open subset of \(\mathcal{A}\), called the elliptic locus, where the general linear sections can be chosen to lie entirely. In those cases, the perverse Leray filtration is shown to be compatible with cup products and the result follows. However, not all cases are handled as such, and so they address the exceptions using an ad hoc argument.


14D20 Algebraic moduli problems, moduli of vector bundles
53D30 Symplectic structures of moduli spaces
Full Text: DOI arXiv


[1] A. B. Altman, A. Iarrobino, and S. L. Kleiman, ”Irreducibility of the compactified Jacobian,” in Real and Complex Singularities, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 1-12. · Zbl 0415.14014
[2] A. B. Altman and S. L. Kleiman, ”Compactifying the Picard scheme,” Adv. in Math., vol. 35, iss. 1, pp. 50-112, 1980. · Zbl 0427.14015 · doi:10.1016/0001-8708(80)90043-2
[3] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves. Vol. I, New York: Springer-Verlag, 1985, vol. 267. · Zbl 0559.14017
[4] V. I. Arnol\('\)d, S. M. Guseuin-Zade, and A. N. Varchenko, Singularities of Differentiable Maps. Vol. II: Monodromy and Asymptotics of Integrals, Boston, MA: Birkhäuser, 1988, vol. 83. · Zbl 0659.58002
[5] A. Beauville, ”Counting rational curves on \(K3\) surfaces,” Duke Math. J., vol. 97, iss. 1, pp. 99-108, 1999. · Zbl 0999.14018 · doi:10.1215/S0012-7094-99-09704-1
[6] A. Beauville, M. S. Narasimhan, and S. Ramanan, ”Spectral curves and the generalised theta divisor,” J. Reine Angew. Math., vol. 398, pp. 169-179, 1989. · Zbl 0666.14015 · doi:10.1515/crll.1989.398.169
[7] A. A. Beuilinson, J. Bernstein, and P. Deligne, ”Faisceaux pervers,” in Analysis and Topology on Singular Spaces, I, Paris: Soc. Math. France, 1982, vol. 100, pp. 5-171. · Zbl 0536.14011
[8] J. D. Copeland, ”Monodromy of the Hitchin map over hyperelliptic curves,” Int. Math. Res. Not., vol. 2005, iss. 29, pp. 1743-1785, 2005. · Zbl 1093.14016 · doi:10.1155/IMRN.2005.1743
[9] K. Corlette, ”Flat \(G\)-bundles with canonical metrics,” J. Differential Geom., vol. 28, iss. 3, pp. 361-382, 1988. · Zbl 0676.58007
[10] A. Corti and M. Hanamura, ”Motivic decomposition and intersection Chow groups. I,” Duke Math. J., vol. 103, iss. 3, pp. 459-522, 2000. · Zbl 1052.14504 · doi:10.1215/S0012-7094-00-10334-1
[11] M. A. A. de Cataldo, ”The perverse filtration and the Lefschetz Hyperplane Section Theorem, II,” J. Algebraic Geom., vol. 12, pp. 305-345. · Zbl 1273.14043 · doi:10.1090/S1056-3911-2011-00566-3
[12] M. A. A. de Cataldo, ”The standard filtration on cohomology with compact supports with an appendix on the base change map and the Lefschetz hyperplane theorem,” in Interactions of Classical and Numerical Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2009, vol. 496, pp. 199-220. · Zbl 1184.14029
[13] M. A. A. de Cataldo and L. Migliorini, The Hodge theory of character varieties. · Zbl 1321.14012
[14] M. A. A. de Cataldo and L. Migliorini, Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces. · Zbl 1304.14011 · doi:10.5427/jsing.2013.7c
[15] M. A. A. de Cataldo and L. Migliorini, ”The Hodge theory of algebraic maps,” Ann. Sci. École Norm. Sup., vol. 38, iss. 5, pp. 693-750, 2005. · Zbl 1094.14005 · doi:10.1016/j.ansens.2005.07.001
[16] M. A. A. de Cataldo and L. Migliorini, ”Hodge-theoretic aspects of the decomposition theorem,” in Algebraic Geometry-Seattle 2005. Part 2, Providence, RI: Amer. Math. Soc., 2009, pp. 489-504. · Zbl 1207.14016
[17] M. A. A. de Cataldo and L. Migliorini, ”The perverse filtration and the Lefschetz hyperplane theorem,” Ann. of Math., vol. 171, iss. 3, pp. 2089-2113, 2010. · Zbl 1213.14017 · doi:10.4007/annals.2010.171.2089
[18] M. A. A. de Cataldo and L. Migliorini, ”The decomposition theorem, perverse sheaves and the topology of algebraic maps,” Bull. Amer. Math. Soc., vol. 46, iss. 4, pp. 535-633, 2009. · Zbl 1181.14001 · doi:10.1090/S0273-0979-09-01260-9
[19] W. Y. Chuang, D. E. Diaconescu, and G. Pan, BPS states and the \(P=W\) conjecture. · Zbl 1320.14057
[20] P. Deligne, ”Décompositions dans la catégorie dérivée,” in Motives, Providence, RI: Amer. Math. Soc., 1994, vol. 55, pp. 115-128. · Zbl 0809.18008
[21] A. Dimca, Sheaves in Topology, New York: Springer-Verlag, 2004. · Zbl 1043.14003
[22] S. K. Donaldson, ”Twisted harmonic maps and the self-duality equations,” Proc. London Math. Soc., vol. 55, iss. 1, pp. 127-131, 1987. · Zbl 0634.53046 · doi:10.1112/plms/s3-55.1.127
[23] C. D’Souza, ”Compactification of generalised Jacobians,” Proc. Indian Acad. Sci. Sect. A Math. Sci., vol. 88, iss. 5, pp. 419-457, 1979. · Zbl 0442.14016
[24] R. Donagi and E. Markman, ”Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles,” in Integrable Systems and Quantum Groups, New York: Springer-Verlag, 1996, vol. 1620, pp. 1-119. · Zbl 0853.35100 · doi:10.1007/BFb0094792
[25] W. Fulton, Intersection Theory, Second ed., New York: Springer-Verlag, 1998, vol. 2. · Zbl 0885.14002
[26] M. Gagne, Compactified Jacobians of Integral Curves with Double Points, ProQuest LLC, Ann Arbor, MI, 1997.
[27] A. Grothendieck, ”Sur le mémoire de Weil: généralisation des fonctions abéliennes,” in Séminaire Bourbaki, Vol. 4, Paris: Soc. Math. France, 1995, p. exp. no. 141, 57-71.
[28] T. Hausel, ”Compactification of moduli of Higgs bundles,” J. Reine Angew. Math., vol. 503, pp. 169-192, 1998. · Zbl 0930.14016 · doi:10.1515/crll.1998.096
[29] T. Hausel, ”Vanishing of intersection numbers on the moduli space of Higgs bundles,” Adv. Theor. Math. Phys., vol. 2, iss. 5, pp. 1011-1040, 1998. · Zbl 1036.81510
[30] T. Hausel, ”Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve,” in Geometric methods in algebra and number theory, Boston, MA: Birkhäuser, 2005, pp. 193-217. · Zbl 1099.14026 · doi:10.1007/0-8176-4417-2_9
[31] T. Hausel, Global topology of the Hitchin system. · Zbl 1322.14027
[32] T. Hausel and C. Pauly, Prym varieties of spectral covers. · Zbl 1264.14061 · doi:10.2140/gt.2012.16.1609
[33] T. Hausel and M. Thaddeus, ”Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles,” J. Amer. Math. Soc., vol. 16, iss. 2, pp. 303-327, 2003. · Zbl 1015.14018 · doi:10.1090/S0894-0347-02-00417-4
[34] T. Hausel and M. Thaddeus, ”Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles,” Proc. London Math. Soc., vol. 88, iss. 3, pp. 632-658, 2004. · Zbl 1060.14048 · doi:10.1112/S0024611503014618
[35] T. Hausel and F. Rodriguez-Villegas, ”Mixed Hodge polynomials of character varieties,” Invent. Math., vol. 174, iss. 3, pp. 555-624, 2008. · Zbl 1213.14020 · doi:10.1007/s00222-008-0142-x
[36] N. Hitchin, ”The self-duality equations on a Riemann surface,” Proc. London Math. Soc., vol. 55, iss. 1, pp. 59-126, 1987. · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59
[37] N. Hitchin, ”Stable bundles and integrable systems,” Duke Math. J., vol. 54, iss. 1, pp. 91-114, 1987. · Zbl 0627.14024 · doi:10.1215/S0012-7094-87-05408-1
[38] B. Iversen, Cohomology of Sheaves, New York: Springer-Verlag, 1986. · Zbl 0559.55001
[39] M. Kashiwara and P. Schapira, Sheaves on Manifolds, New York: Springer-Verlag, 1990, vol. 292. · Zbl 0709.18001
[40] G. Laumon and B. C. Ngô, ”Le lemme fondamental pour les groupes unitaires,” Ann. of Math., vol. 168, iss. 2, pp. 477-573, 2008. · Zbl 1179.22019 · doi:10.4007/annals.2008.168.477
[41] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford: Oxford University Press, 2002, vol. 6. · Zbl 0996.14005
[42] M. Mereb, On the E-polynomials of a family of Character Varieties, 2010. · Zbl 1333.14041 · doi:10.1007/s00208-015-1183-2
[43] M. S. Narasimhan and C. S. Seshadri, ”Stable and unitary vector bundles on a compact Riemann surface,” Ann. of Math., vol. 82, pp. 540-567, 1965. · Zbl 0171.04803 · doi:10.2307/1970710
[44] P. E. Newstead, ”Characteristic classes of stable bundles of rank \(2\) over an algebraic curve,” Trans. Amer. Math. Soc., vol. 169, pp. 337-345, 1972. · Zbl 0256.14008 · doi:10.2307/1996247
[45] N. Nitsure, ”Moduli space of semistable pairs on a curve,” Proc. London Math. Soc., vol. 62, iss. 2, pp. 275-300, 1991. · Zbl 0733.14005 · doi:10.1112/plms/s3-62.2.275
[46] B. C. Ngô, ”Fibration de Hitchin et endoscopie,” Invent. Math., vol. 164, iss. 2, pp. 399-453, 2006. · Zbl 1098.14023 · doi:10.1007/s00222-005-0483-7
[47] B. C. Ngô, ”Le lemme fondamental pour les algèbres de Lie,” Publ. Math. Inst. Hautes Études Sci., iss. 111, pp. 1-169, 2010. · Zbl 1200.22011 · doi:10.1007/s10240-010-0026-7
[48] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, New York: Springer-Verlag, 2008. · Zbl 1138.14002 · doi:10.1007/978-3-540-77017-6
[49] C. J. Rego, ”The compactified Jacobian,” Ann. Sci. École Norm. Sup., vol. 13, iss. 2, pp. 211-223, 1980. · Zbl 0478.14024
[50] C. T. Simpson, ”The ubiquity of variations of Hodge structure,” in Complex Ceometry and Lie Theory, Providence, RI: Amer. Math. Soc., 1991, vol. 53, pp. 329-348. · Zbl 0770.14014
[51] C. T. Simpson, ”Higgs bundles and local systems,” Inst. Hautes Études Sci. Publ. Math., iss. 75, pp. 5-95, 1992. · Zbl 0814.32003 · doi:10.1007/BF02699491
[52] M. Thaddeus, Topology of the moduli space of stable vector bundles over a compact Riemann surface, 1989.
[53] A. Weil, ”Généralisation des fonctions abéliennes,” J. Math. Pures Appl., vol. 17, pp. 47-87, 1938. · JFM 64.0361.02
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