Type II almost-homogeneous manifolds of cohomogeneity one

Guan, Daniel

doi:10.2140/pjm.2011.253.383

Cited by 4 publications

(11 citation statements)

References 13 publications

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“…In [1], we prove the following: Proposition 1. For any simply connected Type I, compact, Kähler, complex, almost-homogeneous manifold of cohomogeneity one with a hypersurface end, there is an extremal metric in a given Kähler class if and only if Condition (7) in [1] holds.…”

Section: Introductionmentioning

confidence: 92%

“…Now, we apply our arguments in [1][2][3]8]. In this note, we only consider the case in which S = SO(n + 1, C).…”

Section: Preliminariesmentioning

confidence: 99%

“…From the proof of Lemma 6 in [1], we see that the eigenvalues of the Ricci curvature of the exceptional divisor in the direction other than the 1-strings of roots with zero eigenvalues are represented as…”

Section: Preliminariesmentioning

confidence: 99%

“…The authors in [10] studied a few of the first cases in [1], i.e., the cases with F = F(OP n ) in which S = SO(4, C), SO(6, C), SO(8, C), SO(10, C), where S is the induced group action on the fibers.…”

Section: Introductionmentioning

confidence: 99%

“…In [1], we used very explicit and elementary calculations to avoid the Cauchy-Riemann structure and other very abstract tools in [10,20] that also cumulating results from other papers of Spiro.…”

Section: Introductionmentioning

confidence: 99%

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Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Guan

Orellana

Van³

2018

Axioms

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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. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m ′ when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.

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Section: Introductionmentioning

confidence: 92%

“…Now, we apply our arguments in [1][2][3]8]. In this note, we only consider the case in which S = SO(n + 1, C).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Guan

Orellana

Van³

2018

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Type II compact almost homogeneous manifolds of cohomogeneity one-II

Guan

2019

Int. J. Math.

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Type I almost homogeneous manifolds of cohomogeneity one, III

Guan¹

2013

Pacific J. Math.

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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.

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Type II almost-homogeneous manifolds of cohomogeneity one

Cited by 4 publications

References 13 publications

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Type II compact almost homogeneous manifolds of cohomogeneity one-II

Type I almost homogeneous manifolds of cohomogeneity one, III

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