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On Kähler nilmanifolds with top homology in codimension two

Published online by Cambridge University Press:  17 April 2009

Bruce Gilligan
Bruce Gilligan
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, Canada, S4S 0A2, e-mail: gilligan@math.uregina.ca
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Suppose G is a connected, complex, nilpotent Lie group and Γ is a discrete subgroup of G such that G/Γ is Kähler and the top nonvanishing homology group of G/Γ (with coefficients in ℤ2) is in codimension two or less. We show that G is then Abelian. We also note that an example from [12] shows that this fails if the top nonvanishing homology is in codimension three.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 1 , August 2006 , pp. 85 - 90
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Akhiezer, D.N. and Gilligan, B., ‘On complex homogeneous spaces with top homology in codimension two’, Canad. J. Math. 46 (1994), 897–919.CrossRefGoogle Scholar
[2]Auslander, L. and Tolimieri, R., ‘Splitting theorems and the structure of solvmanifolds’, Ann. of Math. 2 92 (1970), 164–173.CrossRefGoogle Scholar
[3]Berteloot, F., ‘Existence d'une structure kählérienne sur les variétés homogènes semi-simples’, C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), 809–812.Google Scholar
[4]Berteloot, F. and Oeljeklaus, K., ‘Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups’, Math. Ann. 281 (1988), 513–530.CrossRefGoogle Scholar
[5]Borel, A. and Remmert, R., ‘Über kompakte homogene Kählersche Mannigfaltigkeiten’, Math. Ann. 145 (1962), 429–439.CrossRefGoogle Scholar
[6]Dorfmeister, J. and Nakajima, K., ‘The fundamental conjecture for homogeneous Kähler manifolds’, Acta Math. 161 (1988), 23–70.CrossRefGoogle Scholar
[7]Gilligan, B. and Huckleberry, A.T., ‘On non-compact complex nilmanifolds’, Math. Ann. 238 (1978), 39–49.CrossRefGoogle Scholar
[8]Huckleberry, A.T. and Livorni, E.L., ‘A classification of homogeneous surfaces’, Canad. J. Math. 33 (1981), 1097–1110.CrossRefGoogle Scholar
[9]Matsushima, Y., ‘Sur les espaces homogènes kählériens d'un groupe de Lie réductif’, Nagoya Math. J. 11 (1957), 53–60.CrossRefGoogle Scholar
[10]Milnor, J., Morse theory (Princeton University Press, Princeton, 1963).CrossRefGoogle Scholar
[11]Mostow, G.D., ‘Some applications of representative functions to solvmanifolds’, Amer. J. Math. 93 (1971), 11–32.CrossRefGoogle Scholar
[12]Oeljeklaus, K. and Richthofer, W., ‘On the structure of complex solvmanifolds’, J. Differential Geom. 27 (1988), 399–421.CrossRefGoogle Scholar