Preprint Article Version 1 This version is not peer-reviewed

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Version 1 : Received: 16 November 2018 / Approved: 19 November 2018 / Online: 19 November 2018 (12:04:09 CET)

A peer-reviewed article of this Preprint also exists.

Guan, Z.-D.D.; Orellana, P.; Van, A. Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV. Axioms 2019, 8, 2. Guan, Z.-D.D.; Orellana, P.; Van, A. Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV. Axioms 2019, 8, 2.

Journal reference: Axioms 2018, 8, 2
DOI: 10.3390/axioms8010002

Abstract

This is the fourth part of [6] on the existence of K¨ahler Einstein metrics of the general type I almost homogeneous manifolds of cohomogeneity one. We actually carry out all the results in [8] to the type I cases. In part II [14], we obtained a lot of new K¨ahler-Einstein manifolds as well as Fano manifolds without K¨ahler-Einstein metrics. In particular, by applying Theorem 15 therein, we have complete results in the Theorems 3 and 4 in that paper. However, we only have some partial results in Theorem 5 there. In this note, we shall give a report of recent progress on the Fano manifolds Nn,m when n > 15 and N′n,m when n > 4. We actually give two nice pictures for these two classes of manifolds. See our Theorems 1 and 2 in the last section. Moreover, we post two conjectures. Once we could solve these two conjectures, the question for these two classes of manifolds would be completely solved. With applying our results to the canonical circle bundles we also obtain Sasakian manifolds with or without Sasakian-Einstein metrics. That also give some open Calabi-Yau manifolds.

Subject Areas

Kahler manifolds, Einstein metrics, Ricci curvature, fibration, almost-homogeneous, cohomogeneity one, semisimple Lie group, Sasakian Einstein, Calabi-Yau metrics

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