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Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit II

  • Dmitri Alekseevsky
  • Fabio ZuddasEmail author
Article
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Abstract

Podestà and Spiro (Osaka J Math 36(4):805–833, 1999) introduced a class of G-manifolds M with a cohomogeneity one action of a compact semisimple Lie group G which admit an invariant Kähler structure (gJ) (“standard G-manifolds”) and studied invariant Kähler and Kähler–Einstein metrics on M. In the first part of this paper, we gave a combinatoric description of the standard non-compact G-manifolds as the total space \(M_{\varphi }\) of the homogeneous vector bundle \(M = G\times _H V \rightarrow S_0 =G/H\) over a flag manifold \(S_0\) and we gave necessary and sufficient conditions for the existence of an invariant Kähler–Einstein metric g on such manifolds M in terms of the existence of an interval in the T-Weyl chamber of the flag manifold \(F = G \times _H PV\) which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group \(G = SU_n, Sp_n. Spin_n\) and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold \(S_0 = G/H\). If this condition is fulfilled, the explicit construction of the Kähler–Einstein metric reduces to the calculation of the inverse function to a given function of one variable.

Keywords

Kähler–Einstein metrics Cohomogeneity one manifolds Homogeneous vector bundles Flag manifolds Dynkin diagrams 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for her/his very careful comments which helped to improve the exposition of the paper.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.A.A.Kharkevich Institute for Information Transition ProblemsMoscowRussia
  2. 2.Faculty of Science University of Hradec KraloveHradec KraloveCzech Republic
  3. 3.Dipartimento di Matematica e InformaticaCagliariItaly

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