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Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type. (English) Zbl 1159.22006

This paper studies the deformations of compact complex hyperbolic manifolds of complex dimension \(m > 1\) inside Hermitian symmetric spaces of noncompact type. That is, let \(\Gamma\) be a torsionfree cocompact (uniform) lattice in \(SU(m,1)\), and let \(G\) be \(SU(p,2)\) with \(p>2\), \(Sp(2,\mathbb R)\), or \(SO(p,2)\) with \(p\geq 3\). Associated to a representation \(\rho:\Gamma\longrightarrow G\) is the Toledo invariant, \(\tau(\rho)\) which is a discrete scalar invariant, satisfying a bound generalizing the Milnor-Wood inequality. Furthermore \(\tau(\rho)\) is maximized for the representation \(\Gamma\longrightarrow G\) obtained by composing the inclusion \(\Gamma\hookrightarrow SU(m,1)\) with an embedding \(SU(m,1)\hookrightarrow G\) corresponding to a totally geodesic holomorphic embedding \(H^m_{\mathbb C} \hookrightarrow G/K\). Here the symmetric space \(G/K\) is a classical bounded symmetric domain of rank \(2\).
The main result is a converse: \(\tau(\rho)\) is maximized if and only if \(\rho\) is conjugate to such an embedding. This result extends the case when \(G=SU(n,1)\) (D. Toledo [J. Differ. Geom. 29, No. 1, 125–133 (1989; Zbl 0676.57012)] and W. M. Goldman [Discontinuous groups and the Euler class, Thesis, Univ. of California (1980)] when \(n=1\)); the analogous question for other unitary groups \(U(p,q)\) had been established by S. B. Bradlow, O. García-Prada and P. B. Gothen [J. Differ. Geom. 64, No. 1, 111–170 (2003; Zbl 1070.53054)] and M. Burger, A. Iozzi and A. Wienhard [C. R., Math., Acad. Sci. Paris 336, No. 5, 387–390 (2003; Zbl 1035.32013)]. In particular a maximal representation in this case corresponds to an embedding \(SU(p,2) \hookrightarrow G\) (where \(p\geq 2m\)) and \(\rho\) is a discrete reductive embedding stabilizing an embedding of \(H^m_{\mathbb C} \hookrightarrow SU(p,2)/S(U(p)\times U(2))\) as holomorphic totally geodesic submanifold (of maximum possible holomorphic sectional curvature) in \(G/K\).
The methods use Higgs bundle techniques similar to [Bradlow, García-Prada and Gothen (op. cit.); Geom. Dedicata 122, 185–213 (2006; Zbl 1132.14029)], first pioneered by K. Corlette [J. Differ. Geom. 28, No. 3, 361–382 (1988; Zbl 0676.58007], N. J. Hitchin [Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987; Zbl 0634.53045); Topology 31, No. 3, 449–473 (1992; Zbl 0769.32008)] and C. T. Simpson [J. Am. Math. Soc. 1, No. 4, 867–918 (1988; Zbl 0669.58008); Publ. Math., Inst. Hautes Étud. Sci. 75, 5–95 (1992; Zbl 0814.32003)]. Results in the non-uniform case had been proved earlier by the authors in [Ann. Inst. Fourier 58, No. 2, 507–558 (2008; Zbl 1147.22009)].

MSC:

22E40 Discrete subgroups of Lie groups
53C24 Rigidity results
53C35 Differential geometry of symmetric spaces
58E20 Harmonic maps, etc.
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References:

[1] Bradlow S.B., Garcia-Prada O., Gothen P.B.: Surface group representations and U(p,q)-Higgs bundles. J. Diff. Geom. 64, 111–170 (2003) · Zbl 1070.53054
[2] Bradlow S.B., Garcia-Prada O., Gothen P.B.: Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces. Geom. Dedicata. 122, 185–213 (2006) · Zbl 1132.14029 · doi:10.1007/s10711-007-9127-y
[3] Burger M., Iozzi A.: Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity. Geom. Dedicata. 125, 1–23 (2007) · Zbl 1134.53020 · doi:10.1007/s10711-006-9108-6
[4] Burger M., Iozzi A., Wienhard A.: Surface group representations with maximal Toledo invariant. C.R. Acad. Sci. Paris 336, 387–390 (2003) · Zbl 1035.32013
[5] Burger, M., Iozzi, A., Wienhard, A.: Surface group representations with maximal Toledo invariant. arXiv:math.DG/0605656 v2 (2006) · Zbl 1035.32013
[6] Clerc J.-L., Ørsted B.: The Maslov index revisited. Tranform. Groups. 6, 303–320 (2001) · Zbl 1078.53076 · doi:10.1007/BF01237249
[7] Corlette K.: Flat G-bundles with canonical metrics. J. Diff. Geom. 28, 361–382 (1988) · Zbl 0676.58007
[8] Domic A., Toledo D.: The Gromov norm of the Kähler class of symmetric domains. Math. Ann. 276, 425–432 (1987) · Zbl 0595.53061 · doi:10.1007/BF01450839
[9] Eberlein, P.: Geometry of nonpositively curved manifolds. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1996) · Zbl 0883.53003
[10] Goldman, W.M.: Discontinuous groups and the Euler class. Thesis, University of California at Berkeley (1980)
[11] Goldman W.M.: Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988) · Zbl 0655.57019 · doi:10.1007/BF01410200
[12] Goldman W.M., Millson J.J.: Local rigidity of discrete groups acting on complex hyperbolic space. Invent. Math. 88, 495–520 (1987) · Zbl 0627.22012 · doi:10.1007/BF01391829
[13] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original, Graduate Studies in Mathematics, 34. American Mathematical Society, Providence (2001) · Zbl 0993.53002
[14] Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lon. Math. Soc. 55(3), 59–126 (1987) · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59
[15] Hitchin N.J.: Lie groups and Teichmüller space. Topology 31, 449–473 (1992) · Zbl 0769.32008 · doi:10.1016/0040-9383(92)90044-I
[16] Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. In: Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig (1997) · Zbl 0872.14002
[17] Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Princeton University Press (1987) · Zbl 0708.53002
[18] Koziarz V., Maubon J.: Harmonic maps and representations of non-uniform lattices of PU(m,1). Ann. Inst. Fourier (Grenoble) 58(2), 507–558 (2008) · Zbl 1147.22009
[19] Labourie F.: Existence d’applications harmoniques tordues à valeurs dans les variétés à courbure négative. Proc. Am. Math. Soc. 111(3), 877–882 (1991) · Zbl 0783.58016
[20] Royden H.L.: The Ahlfors-Schwarz lemma in several complex variables. Comment. Math. Helvetici. 55, 547–558 (1980) · Zbl 0484.53053 · doi:10.1007/BF02566705
[21] Sampson J.H.: Applications of harmonic maps to Kähler geometry. Contemp. Math. 49, 125–133 (1986) · Zbl 0605.58019
[22] Satake, I.: Algebraic structures of symmetric domains. In: Kanô Memorial Lectures, 4, Iwanami Shoten, Tokyo. Princeton University Press, Princeton (1980) · Zbl 0483.32017
[23] Simpson C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988) · Zbl 0669.58008 · doi:10.1090/S0894-0347-1988-0944577-9
[24] Simpson C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992) · Zbl 0814.32003 · doi:10.1007/BF02699491
[25] Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes Études Sci. Publ. Math. 79, 47–129 (1994) · Zbl 0891.14005 · doi:10.1007/BF02698887
[26] Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety II. Inst. Hautes Études Sci. Publ. Math. 80, 5–79 (1994) · Zbl 0891.14006 · doi:10.1007/BF02698895
[27] Toledo D.: Harmonic mappings of surfaces to certain Kähler manifolds. Math. Scand. 45, 13–26 (1979) · Zbl 0435.58008
[28] Toledo D.: Representations of surface groups in complex hyperbolic space. J. Diff. Geom. 29, 125–133 (1989) · Zbl 0676.57012
[29] Viehweg E., Zuo K.: Arakelov inequalities and the uniformization of certain rigid Shimura varieties. J. Diff. Geom. 77(2), 291–352 (2007) · Zbl 1133.14010
[30] Xia E.Z.: The moduli of flat PU(2,1) structures over Riemann surfaces. Pacific J. Math. 193, 231–256 (2000) · Zbl 1014.32010 · doi:10.2140/pjm.2000.195.231
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