Type I almost homogeneous manifolds of cohomogeneity one, III

Abstract

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. As requested by earlier referees of this series of papers, in this third part, we shall first give an updated description of the geodesic principles and the classification of compact almost homogeneous Kähler manifolds of cohomogeneity one. Then, we shall give a proof of the equivalence of the geodesic stability and the negativity of the integral in the first part. Finally, we shall address the relation of our result to Ross-Thomas version of Donaldson's K-stability. One should easily see that their result is a partial generalization of our integral condition in the first part. And we shall give some further comments on the Fano manifolds with the Ricci classes. In Theorem 14, we give a result of Nadel type. We define the strict slope stability. In our case, it is stronger than Ross-Thomas slope stability. We strengthen two Ross-Thomas results in Theorems 15 and 16. The similar proofs of the results other than the existence for the type II cases are more complicated and will be done elsewhere.

Killing fields generated by multiple solutions to the Fischer–Marsden equation

“…When Guan finished [6] in the Spring 1992, Professor Kobayashi helped him smoothen the language. That eventually led to the solution of cohomogeneity one compact Kähler Einstein manifolds in, for example, [9]. That solution is closely related to the holomorphic vector bundle case presented in [12], see [8].…”

Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics

“…This is motivated by [6]. For example, in the case of cohomogeneity one with a semisimple automorphism group in [7,8], the Futaki invariants are zero. Therefore, there are no Calabi extremal metrics in general.…”

Type II almost-homogeneous manifolds of cohomogeneity one

“…That will also give all the a ρ,s in concrete calculations. This is extremely useful when we check the Fano property of the manifolds and classify the manifolds with higher-codimension ends [Guan 2011a;2011b;≥ 2011b]. For example, we can check this: Proposition 14.…”

“…Not much stability is found there (but see [Guan 2003] for the stability of related constructions). The type I case was dealt with in [Guan 2011a;2011b;≥ 2011b], while the type II case is the subject of this paper and [Guan 2009]. * This is the first class of manifolds for which a criterion for the existence of Calabi extremal metrics has been completely elucidated; it is equivalent to geodesic stability.…”

“…We prove the converse in [Guan ≥ 2011a]. In [Guan 2002;2006;Guan and Chen 2000] we dealt with some examples, and in [Guan 2009] we dealt with the two most conceptually difficult series of manifolds.…”