Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one—II

Guan, Daniel

doi:10.1007/bf02930860

Cited by 9 publications

(52 citation statements)

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“…As in [12], we may expect that is related to a Kähler metric of constant scalar curvature on the normal line bundle over the hypersurface orbit. Hence, we may apply the method of counting zeros in [10,12] to this circumstance. −1 Q 1 ( ) is proportional to " Q" in [10].…”

Section: −1mentioning

confidence: 98%

“…By choosing A( 0 ) close to ( + 2B) we have that (1) = ( + 2B). Arguing as in [12], we have that (1) exists and is finite. Similarly, (0) and (1) exist and are finite.…”

Section: Corollary 82mentioning

confidence: 98%

“…We could easily argue as in [14, p. 273-274] and [12] that the right side of (10) is the Ding-Tian generalized Futaki invariant for a (possibly singular) completion of the normal line bundle of the exceptional divisor, although we do not really know that there is an actually analytic degeneration with this completion as the central fiber. Our condition here is stronger than the Ross-Thomas version of Donaldson's version of K-stability (Cf.…”

Section: There Is a Kähler Metric Of Constant Scalar Curvature In A Gmentioning

confidence: 98%

“…

Abstract: This paper is one in a series generalizing our results in [12,14,15,20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends.

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confidence: 98%

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Affine compact almost-homogeneous manifolds of cohomogeneity one

Guan

2009

Open Mathematics

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This paper is one in a series generalizing our results in [12,14,15,20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

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Section: −1mentioning

confidence: 98%

“…By choosing A( 0 ) close to ( + 2B) we have that (1) = ( + 2B). Arguing as in [12], we have that (1) exists and is finite. Similarly, (0) and (1) exist and are finite.…”

Section: Corollary 82mentioning

confidence: 98%

Section: There Is a Kähler Metric Of Constant Scalar Curvature In A Gmentioning

confidence: 98%

“…

…”

mentioning

confidence: 98%

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Affine compact almost-homogeneous manifolds of cohomogeneity one

Guan

2009

Open Mathematics

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“…There are three types of manifolds of this kind (see [Guan 2002] for details). Type III compact complex almost-homogeneous manifolds of real cohomogeneity one were dealt with in [Guan 1995].…”

Section: Introductionmentioning

confidence: 99%

Type II almost-homogeneous manifolds of cohomogeneity one

Guan¹

2011

Pacific J. Math.

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This paper continues our investigation on the existence of extremal metrics of the general affine and type II almost-homogeneous manifolds of cohomogeneity one. It deals with the general type II cases with hypersurface ends: more precisely, with manifolds having certain ‫ރ‬ P n ‫ރ(×‬ P n ) * -or ‫ރ‬ P 2 -bundle structures. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics.

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