zbMATH — the first resource for mathematics

Harmonic mappings of Kähler manifolds to locally symmetric spaces. (English) Zbl 0695.58010
Define the following extension of Gromov ordering for manifolds M and N not necessarily of the same dimension: \(M\geq N\) means the existence of a continuous map f: \(M\to N\) which is surjective in homology. Now let M be a compact Kähler manifold and N be a compact locally symmetric space of the form \(\Gamma\) \(\setminus G/K\), where G is semisimple Lie group with compact factors, K is a maximal compact subgroup and \(\Gamma\) is a cocompact discrete subgroup. The main result is to show that \(M\geq N\) is impossible unless N is already Kähler in an obvious way, i.e., is locally Hermitian symmetric.
Reviewer: U.D’Ambrosio

58E20 Harmonic maps, etc.
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58A14 Hodge theory in global analysis
53C35 Differential geometry of symmetric spaces
Full Text: DOI Numdam EuDML
[1] D. Barlet, Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie, Seminaire Norguet 1974–1975,Springer Lecture Notes in Mathematics,482 (1975). 1–158.
[2] A. L. Besse,Manifolds all of whose Geodesics are Closed, Springer-Verlag, 1978. · Zbl 0387.53010
[3] E. Bishop, Conditions for the analyticity of certain sets,Mich. Math. J.,11 (1964), 289–304. · Zbl 0143.30302
[4] R. Bott andH. Samelson, Applications of the theory of Morse to symmetric spaces,Amer. Jour. Math.,80 (1958), 964–1029. · Zbl 0101.39702
[5] J. A. Carlson, Bounds on the dimension of a variation of Hodge structure,Trans. A.M.S.,294 (1986), 45–64, · Zbl 0593.14006
[6] J. A. Carlson andD. Toledo, Variations of Hodge structure, Legendre submanifolds and accessibility,Trans. A.M.S., 311 (1989), 391–411. · Zbl 0667.14004
[7] R. Charney andR. Lee, Characteristic classes for the classifying spaces of Hodge structures,K-Theory,1 (1987), 237–270. · Zbl 0647.14003
[8] K. Corlette, Flat G-bundles with canonical metrics,J. Diff. Geom.,28 (1988), 361–382. · Zbl 0676.58007
[9] M. Cornalba andP. A. Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties,Invent. Math.,28 (1975), 1–106. · Zbl 0293.32026
[10] P. Deligne, P. A. Griffiths, J. Morgan andD. Sullivan, Real homotopy theory of Kähler manifolds,Invent. Math.,29 (1975), 245–274. · Zbl 0312.55011
[11] J. Eells andS. Salamon, Constructions twistorielles des applications harmoniques,C.R. Acad. Sci. Paris, I,296 (1983), 685–687. · Zbl 0531.58020
[12] J. Eells andJ. H. Sampson, Harmonic mappings of Riemannian manifolds,Amer. Jour. Math.,86 (1964), 109–160. · Zbl 0122.40102
[13] P. A. Griffiths, Periods of integrals on algebraic manifolds, III,Publ. Math. I.H.E.S.,38 (1970), 125–180. · Zbl 0212.53503
[14] P. A. Griffiths andW. Schmid, Locally homogeneous complex manifolds,Acta Math.,123 (1969), 253–302. · Zbl 0209.25701
[15] S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978. · Zbl 0451.53038
[16] J. Jost andS. T. Yau, Harmonic mappings and Kähler manifolds,Math. Ann.,262 (1983), 145–166. · Zbl 0527.53041
[17] J. Jost andS. T. Yau,Harmonic maps and group representations, to appear. · Zbl 0729.58024
[18] D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups,Funct. Anal. Appl.,1 (1967), 63–65. · Zbl 0168.27602
[19] B. Kostant andS. Rallis, Orbits and representations associated with symmetric spaces,Amer. Jour. Math.,93 (1971), 753–809. · Zbl 0224.22013
[20] D. I. Liebermann, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Seminaire Norguet 1975–1977,Springer Lecture Notes in Mathematics,670 (1978), 140–186.
[21] G. A. Margulis, Discrete groups of motions of manifolds with non-positive curvature (in Russian),Proc. Int. Cong. Math., Vancouver, 1974, vol. 2, 35–44.
[22] N. Mok, The holomorphic or anti-holomorphic character of harmonic maps into irreducible compact quotients of polydiscs,Math. Ann.,272 (1985), 197–216. · Zbl 0604.58021
[23] J. W. Morgan, The algebraic topology of smooth algebraic varieties,Publ. Math. I.H.E.S.,48 (1978), 137–204. · Zbl 0401.14003
[24] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces,Annals of Math. Studies,78, Princeton University Press, 1973. · Zbl 0265.53039
[25] S. Salamon, Harmonic and holomorphic maps,Springer Lecture Notes in Mathematics,1164, 161–224. · Zbl 0591.53031
[26] J. H. Sampson, Some properties and applications of harmonic mappings,Ann. Sci. École Norm. Sup.,11 (1978), 211–228. · Zbl 0392.31009
[27] J. H. Sampson, Applications of harmonic maps to Kähler geometry,Contemp. Math.,49 (1986), 125–133. · Zbl 0605.58019
[28] Y. T. Siu, Complex analyticity of harmonic maps and strong rigidity of complex Kähler manifolds,Ann. Math.,112 (1980), 73–111. · Zbl 0517.53058
[29] Y. T. Siu, Strong rigidity of compact quotients of exceptional bounded symmetric domains,Duke Math. Jour.,48 (1981), 857–871. · Zbl 0496.32020
[30] Y. T. Siu, Complex analyticity of harmonic maps, vanishing and Lefschetz theorems,J. Diff. Geom.,17 (1982), 55–138. · Zbl 0497.32025
[31] Y. T. Siu, Strong rigidity for Kähler manifolds and the construction of bounded holomorphic functions, inDiscrete Groups in Geometry and Analysis,R. Howe (editor), Birkhauser, 1987, 124–151.
[32] R. J. Zimmer, Kazhdan groups acting on compact manifolds,Invent. Math.,75 (1984), 425–436. · Zbl 0576.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.