Affine compact almost-homogeneous manifolds of cohomogeneity one

Guan, Daniel

doi:10.2478/s11533-008-0055-3

Cited by 3 publications

(18 citation statements)

References 20 publications

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“…If we assume simple connectedness, such manifolds are automatically projective. It is interesting to ask when they are Fano, Kähler-Einstein, and so on [Guan 2009]. …”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, to finish the affine case and the type II case, we have to deal with this case. Individually, these manifolds seem easier to deal with than those in [Guan 2009]. However, there are more of them, and it turns out that as a group and analytically, they are technically more involved.…”

Section: Introductionmentioning

confidence: 99%

“…(One could also consider the path as generated by a semisimple element H α , where α is the root that generates the sl(2) Lie algebra Ꮽ.) Then we apply the same argument given in Section 3 of [Guan 2006;2009] to the affine A n -action case. We find that the same method works for the complex structure of both the affine and the type II cases.…”

Section: Introductionmentioning

confidence: 99%

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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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“…If we assume simple connectedness, such manifolds are automatically projective. It is interesting to ask when they are Fano, Kähler-Einstein, and so on [Guan 2009]. …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Type II almost-homogeneous manifolds of cohomogeneity one

Guan¹

2011

Pacific J. Math.

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

Guan

2014

Geometry

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There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.

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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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Affine compact almost-homogeneous manifolds of cohomogeneity one

Cited by 3 publications

References 20 publications

Type II almost-homogeneous manifolds of cohomogeneity one

Type II almost-homogeneous manifolds of cohomogeneity one

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

Type I almost homogeneous manifolds of cohomogeneity one, III

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