Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Kähler structures on compact solvmanifolds

Authors: Chal Benson and Carolyn S. Gordon
Journal: Proc. Amer. Math. Soc. 108 (1990), 971-980
MSC: Primary 53C55; Secondary 22E25, 22E40, 32M05, 32M10
MathSciNet review: 993739
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] C. Benson and C. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518. MR 976592 (90b:53042)
  • [2] L. A. Cordero, M. Fernández and A. Gray, Compact symplectic manifolds not admitting positive definite Kähler metrics, preprint.
  • [3] J. Dorfmeister and K. Nakajima, The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), 23-70. MR 962095 (89i:32066)
  • [4] M. Fernández and A. Gray, Compact symplectic solvmanifolds not admitting complex structures, (to appear in J. Geom. Phys.).
  • [5] S. G. Gindikin, I. I. Pjatecckii-Shapiro and E. B. Vinberg, Homogeneous Kähler manifolds, translation by A. Koranyi, in Geometry of homogeneous domains, Edizioni Cremonese, Rome, 1968. MR 0238237 (38:6513)
  • [6] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley and Sons, New York, 1978. MR 507725 (80b:14001)
  • [7] P. Griffiths and J. Morgan, Rational homotopy theory and differential forms, Birkhäuser, Boston, 1981. MR 641551 (82m:55014)
  • [8] J. Hano, On Kählerian homogeneous spaces of unimodular Lie groups, Amer. J. Math., 79 (1957), 885-900. MR 0095979 (20:2477)
  • [9] K. Hasegawa, Minimal models of nilmanifolds, preprint. MR 946638 (89i:32015)
  • [10] A. Hattori, Spectral sequence in the deRham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo, Sect. 1, 8 (1960), 289-331. MR 0124918 (23:A2226)
  • [11] D. McDuff, The momentum map for circle actions on symplectic manifolds, (to appear in J. Geom. Phys.). MR 1029424 (91c:58042)
  • [12] K. Nomizu, On the cohomology of homogeneous spaces of nilpotent Lie groups, Ann. Math. 59 (1954), 531-538. MR 0064057 (16:219c)
  • [13] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer, Berlin and New York, 1972. MR 0507234 (58:22394a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C55, 22E25, 22E40, 32M05, 32M10

Retrieve articles in all journals with MSC: 53C55, 22E25, 22E40, 32M05, 32M10

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society