Abstract
Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevsky, D.: Flag manifolds, 11. Yugoslav Geometrical seminar, Divcibare, 10–17 October, 3–35 (1993)
Alekseevsky, D., Spiro, A.: Invariant CR structures on compact homogeneous manifolds. Hokk. Math. J. 32(2), 209–276 (2003)
Alekseevsky, D.V., Perelomov, A.M.: Invariant Kähler–Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl. 20(3), 171–182 (1986)
Arvanitoyeorgos, A.: Geometry of flag manifolds. Int. J. Geom. Methods Mod. Phys. 3(5, 6), 957–974 (2006)
Besse, A.: Einstein manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin (1987)
Berard- Bergery, L.: Sur des nouvelles varietes Riemanniennes d’Einstein, Publ. de nst. E. Cartan, No. 6, 1–60 (1982)
Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces. Am. J. Math. 80, 458–538 (1958)
Dancer, A., Wang, M.Y.: Kähler Einstein metrics of cohomogeneity one and bundle construction for Einstein Hermitian metrics. Math. Ann. 312, 503–526 (1998)
Eschenburg, J.-H., Wang, M.Y.: The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10(1), 109–137 (2000)
Gorbatsevich, V.V., Onishchik, A.L., Vinberg, E.B.: Structure of Lie groups and Lie algebras, Encycl. Math. Sci., Lie groups and Lie algebras, III, Springer Verlag
Huckleberry, A., Snow, D.: Almost homogeneous Kahler manifolds with hypersurface orbits. Osaka J. Math. 19, 763–786 (1982)
Koiso, N., Sakane, Y.: Non-homogeneous Kähler–Einstein metrics on compact complex manifolds, Curvature and topology of Riemannian manifolds. In: Proc. 17th Int. Taniguchi Symp., Katata/Jap.1985, Lect. Notes Math. 1201, pp. 165–179 (1986)
Koiso, N., Sakane, Y.: Non homogeneous Kähler-Einstein metrics on compact complex manifolds II. Osaka J. Math. 25, 933–959 (1988)
Page, D.N., Pope, C.N.: Inhomogeneous Einstein metrics on complex line bundles. Class. Quantum Gravity 4(2), 213–225 (1987)
Podestà, F., Spiro, A.: Kähler manifolds with large isometry group. Osaka J. Math. 36(4), 805–833 (1999)
Sakane, Y.: Examples of compact Einstein-Kähler manifolds with positive Ricci tensor. Osaka J. Math. 23, 585–616 (1986)
Snow, D.M.: Homogeneous vector bundles, Group actions and invariant theory (Montreal, PQ, 1988). In: CMS Conf. Proc., 10, 193205, Amer. Math. Soc., Providence, RI (1989)
Verdiani, L.: Invariant metrics on cohomogeneity one manifolds. Geometriae Dedicata 77(1), 77–110 (1999)
Acknowledgements
The authors would like to thank Podestà and Spiro for useful discussions, suggestions and clarifications on their results.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alekseevsky, D., Zuddas, F. Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I. Ann Glob Anal Geom 52, 99–128 (2017). https://doi.org/10.1007/s10455-017-9550-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-017-9550-8