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Kähler groups and rigidity phenomena

Published online by Cambridge University Press:  24 October 2008

F. E. A. Johnson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT
E. G. Rees
Affiliation:
Department of Mathematics, University of Edinburgh, The King's Buildings, Edinburgh EH9 3JZ

Extract

The class of fundamental groups of non-singular complex projective varieties is an interesting, but as yet imperfectly understood, class of finitely presented groups. Membership of is known to be extremely restricted (see [22, 23]). In this paper, we employ geometrical rigidity properties to realize some group extensions as elements of as in our previous papers, we find it convenient to work simultaneously with the class ℋ of fundamental groups of compact Kähler manifolds.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Albert, A. A.. Involutorial simple algebras and real Riemann matrices. Ann. of Math. 36 (1935), 886964.CrossRefGoogle Scholar
[2]Atiyah, M. F.. The signature of fibre bundles. In Collected Papers in Honour of K. Kodaira (Tokyo University Press, 1969). pp. 7384.Google Scholar
[3]Auslander, L. and Kuranishi, M.. On the holonomy group of locally Euclidean spaces. Ann. of Math. 65 (1957), 411415.CrossRefGoogle Scholar
[4]Ballmann, W.. Nonpositively curved manifolds of higher rank. Ann. of Math. 122 (1985), 597609.CrossRefGoogle Scholar
[5]Baily, W. L.. Introductory Lectures on Automorphic Forms (Princeton University Press, 1973).Google Scholar
[6]Bieberbach, L.. Über die Bewegungsgruppen der Euklidischen Raüme I. Math. Ann. 70 (1911), 297336.CrossRefGoogle Scholar
[7]Bishop, R. L. and O'Neill, B.. Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 (1969), 149.CrossRefGoogle Scholar
[8]Borel, A.. Density properties of certain subgroups of semisimple groups. Ann. of Math. 72 (1960), 179188.CrossRefGoogle Scholar
[9]Borel, A.. On the automorphisms of certain subgroups of semisimple Lie groups. In Algebraic Geometry. Papers Presented at the Bombay Colloquium, 1968 (Oxford University Press, 1969). pp. 4373.Google Scholar
[10]Charlap, L. S.. Bieberbach Groups and Flat Manifolds (Springer-Verlag, 1986).CrossRefGoogle Scholar
[11]Deligne, P. and Mostow, G. D.. Monodromy of hypergeometric functions and non-lattice integral monodromy. Inst. Hautés Études Sci. Publ. Math. 63 (1986), 589.CrossRefGoogle Scholar
[12]Eberlein, P.. Isometry groups of simply connected manifolds of nonpositive curvature II. Acta Math. 149 (1982), 4169.CrossRefGoogle Scholar
[13]Eells, J. and Sampson, J. H.. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109160.CrossRefGoogle Scholar
[14]Godeaux, L.. Sur une surface algébrique de genre zero et de bigenre deux. Atti Accad. Naz. Lincei 14 (1931), 479481.Google Scholar
[15]Hartman, P.. On homotopic harmonic maps. Canad. J. Math. 19 (1967), 673687.CrossRefGoogle Scholar
[16]Helgason, S.. Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, 1978).Google Scholar
[17]Johnson, F. E. A.. Automorphisms of direct products of groups and their geometric realisations. Math. Ann. 263 (1983), 343364.CrossRefGoogle Scholar
[18]Johnson, F. E. A.. A class of non-Kählerian manifolds. Math. Proc. Cambridge Philos. Soc. 100 (1986), 519521.CrossRefGoogle Scholar
[19]Johnson, F. E. A.. On the existence of irreducible discrete subgroups in isotypic Lie groups of classical type. Proc. London Math. Soc. 56 (1988), 5177.CrossRefGoogle Scholar
[20]Johnson, F. E. A.. Flat algebraic manifolds. In Proceedings of the 1989 Durham Symposium on Low-Dimensional Topology (Cambridge University Press, to appear).Google Scholar
[21]Johnson, F. E. A.. Extending group actions by finite groups. (To appear in Topology.)Google Scholar
[22]Johnson, F. E. A. and Rees, E. G.. On the fundamental group of a complex algebraic manifold. Bull. London Math. Soc. 19 (1987), 463466.CrossRefGoogle Scholar
[23]Johnson, F. E. A. and Rees, E. G.. On the fundamental group of a complex algebraic manifold II. In Proceedings of Poznan International Conference on Algebraic Topology 1989. (To appear.)Google Scholar
[24]Kerckhoff, S. P.. The Nielsen realisation problem. Ann. of Math. (2) 117 (1983), 235265.CrossRefGoogle Scholar
[25]Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry, vol. 2 (Wiley-Interscience, 1969).Google Scholar
[26]Kodaira, K.. On Kähler varieties of restricted type. Ann. of Math. 60 (1954), 2848.CrossRefGoogle Scholar
[27]Kodaira, K.. A certain type of irregular algebraic surface. Jour. Anal. Math. 19 (1967), 207215.CrossRefGoogle Scholar
[28]Koebe, P.. Über die Uniformisierung algebraischer Kurven. Math. Ann. 67 (1909), 145224.CrossRefGoogle Scholar
[29]Lemaire, L.. Harmonic mappings of uniformly bounded dilation. Topology 16 (1977), 199201.CrossRefGoogle Scholar
[30]Lichnerowicz, A.. Applications harmoniques et variétés Kählériennes. Sympos. Math. 3 (1970), 341402.Google Scholar
[31]Maclane, S.. Homology (Springer-Verlag, 1967).Google Scholar
[32]Mal'cev, A. I.. On a class of homogeneous spaces. Amer. Math. Soc. Transl. 39 (American Mathematical Society 1951).Google Scholar
[33]Mangler, W.. Die Klassen topologischer Abbildungen einer geschlossenen Fläche auf sich. Math. Z. 44 (1939), 541554.CrossRefGoogle Scholar
[34]Mostow, G. D.. Strong rigidity of locally symmetric spaces. Annals of Math. Studies no. 78 (Princeton University Press, 1973).Google Scholar
[35]Mostow, G. D. and Siu, Y. T.. A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. 112 (1980), 321360.CrossRefGoogle Scholar
[36]Nielsen, J.. Abbildungsklassen endlicher Ordnung. Acta Math. 75 (1943), 23115.CrossRefGoogle Scholar
[37]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Ergeb. der Math. no. 68 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[38]Serre, J. P.. Sur la topologie des variétés algébriques en caractéristique p. In Symp. Internacional de Topologia Algebraica, Mexico City (Universidad de Mexico, 1958). pp. 2453.Google Scholar
[39]Shafarevitch, I. R.. Basic Algebraic Geometry (Springer-Verlag, 1974).CrossRefGoogle Scholar
[40]Siu, Y. T.. The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math. (2) 112 (1980), 73111.CrossRefGoogle Scholar
[41]Webber, D. J. St. H.. On the existence of irreducible discrete subgroups in isotypic Lie groups of exceptional type. Ph.D. Thesis, University of London (1985).Google Scholar
[42]Whitehead, G. W.. Elements of Homotopy Theory. Graduate Texts in Math. no. 61 (Springer-Verlag, 1978).CrossRefGoogle Scholar