Type II compact almost homogeneous manifolds of cohomogeneity one-II

Guan, Daniel

doi:10.1142/s0129167x19400020

Cited by 2 publications

(14 citation statements)

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“…We obtained similar results for the higher codimensional end case and the general case, in Sections 3 and 4 of Part II [3]. See also Theorem 15 in the last section for the Kähler-Einstein case therein.…”

Section: Introductionsupporting

confidence: 67%

“…We also obtained many Kähler-Einstein manifolds as well as Fano manifolds without the Kähler-Einstein metric of Type I in the fifth section of Part II [3]. The Futaki invariants are zero in our case since the automorphism groups are semisimple.…”

Section: Introductionmentioning

confidence: 92%

“…To understand better the M n in our Section 5 in Part II [3], that is, the M n,1 in our Theorem 3 there, it is not difficult to see that M n is a Q n bundle over…”

Section: Introductionmentioning

confidence: 96%

“…In Part II [3], Theorems 3 and 4, we completely solved the existence of the Kähler-Einstein metric for the two classes of manifolds M n,m and M n.m therein.…”

Section: Introductionmentioning

confidence: 99%

“…A general reader might use [27] as a reference. In Part II [3], Theorem 5, we completely solved the integral problem for N n,m . They are all homogeneous and hence Kähler-Einstein.…”

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: Introductionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 92%

“…To understand better the M n in our Section 5 in Part II [3], that is, the M n,1 in our Theorem 3 there, it is not difficult to see that M n is a Q n bundle over…”

Section: Introductionmentioning

confidence: 96%

“…In Part II [3], Theorems 3 and 4, we completely solved the existence of the Kähler-Einstein metric for the two classes of manifolds M n,m and M n.m therein.…”

Section: Introductionmentioning

confidence: 99%

“…A general reader might use [27] as a reference. In Part II [3], Theorem 5, we completely solved the integral problem for N n,m . They are all homogeneous and hence Kähler-Einstein.…”

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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Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Guan¹,

Orellana²,

Van³

2018

Preprint

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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.

show abstract

Type II compact almost homogeneous manifolds of cohomogeneity one-II

Cited by 2 publications

References 6 publications

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

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Type II compact almost homogeneous manifolds of cohomogeneity one-II

Cited by 2 publications

References 6 publications

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

Type I Almost-Homogeneous Manifolds of Cohomogeneity One&mdash;IV

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Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV