2019/06/25 by Dmitri V. Alekseevsky, Alekseevsky, Dmitri, Fabio Zuddas +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1906.10633
openalex publication_date 2019/06/25 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
F. Podestà and A. Spiro introduced a class of G-manifolds M with a cohomogeneity one action of a compact semisimple Lie group G which admit an invariant Kaehler structure (g,J) (``standard G-manifolds") and studied invariant Kaehler and Kaehler-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 Mφ of the homogeneous vector bundle M = G×H V → S0 =G/H over a flag manifold S0 and we gave necessary and sufficient conditions for the existence of an invariant Kaehler-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 × H PV which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group G = SUn, Spn. Spinn and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold S0 = G/H. If this conditions is fulfilled, the explicit construction of the Kaehler-Einstein metric reduces to the calculation of the inverse function to a given function of one variable.