Kähler structures on compact solvmanifolds

Benson, Chal; Gordon, Carolyn S.

doi:10.1090/S0002-9939-1990-0993739-4

Kähler structures on compact solvmanifolds
HTML articles powered by AMS MathViewer

by Chal Benson and Carolyn S. Gordon PDF

Proc. Amer. Math. Soc. 108 (1990), 971-980 Request permission

Abstract:

In a previous paper, the authors proved that the only compact nilmanifolds $\Gamma \backslash G$ which admit Kähler structures are tori. Here we consider a more general class of homogeneous spaces $\Gamma \backslash G$, where $G$ is a completely solvable Lie group and $\Gamma$ is a cocompact discrete subgroup. Necessary conditions for the existence of a Kähler structure are given in terms of the structure of $G$ and a homogeneous representative $\omega$ of the Kähler class in ${H^2}(\Gamma \backslash G;\mathbb {R})$. These conditions are not sufficient to imply the existence of a Kähler structure. On the other hand, we present examples of such solvmanifolds that have the same cohomology ring as a compact Kähler manifold. We do not know whether some of these solvmanifolds admit Kähler structures.

References

Chal Benson and Carolyn S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. MR 976592, DOI 10.1016/0040-9383(88)90029-8

Compact symplectic manifolds not admitting positive definite Kähler metrics

Josef Dorfmeister and Kazufumi Nakajima, The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), no. 1-2, 23–70. MR 962095, DOI 10.1007/BF02392294

Compact symplectic solvmanifolds not admitting complex structures

S. G. Gindikin, I. I. Pjateckiĭ-Šapiro, and È. B. Vinberg, Homogeneous Kähler manifolds, Geometry of Homogeneous Bounded Domains (C.I.M.E., 3 Ciclo, Urbino, 1967), Edizioni Cremonese, Rome, 1968, pp. 3–87 (German). MR 0238237
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Phillip A. Griffiths and John W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR 641551
Jun-ichi Hano, On Kaehlerian homogeneous spaces of unimodular Lie groups, Amer. J. Math. 79 (1957), 885–900. MR 95979, DOI 10.2307/2372440
Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. MR 946638, DOI 10.1090/S0002-9939-1989-0946638-X
Akio Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331 (1960). MR 124918
Dusa McDuff, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), no. 2, 149–160. MR 1029424, DOI 10.1016/0393-0440(88)90001-0
Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538. MR 64057, DOI 10.2307/1969716
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1