Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends

Guan, Daniel

doi:10.1090/s0002-9947-1995-1285992-6

Cited by 22 publications

(39 citation statements)

References 6 publications

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“…Our result clarifies the third class. Then we show concretely that for the homogeneous cases, our condition (10) holds. This of course should be true, but we just use it as examples.…”

Section: Introductionmentioning

confidence: 81%

“…There always exists an extremal metric in any Kähler class. Recently, we generalized this existence result to a family of metrics, which connects the extremal metric [10] and the generalized quasi-einstein metric [9], called the extremal-soliton metrics in [16]. The existence of the extremal-soliton is the same as the geodesic stability with respect to a generalized Mabuchi functional.…”

Section: Guanmentioning

confidence: 98%

“…If a compact almost-homogeneous Kähler manifold is a completion of a C * bundle over a product of a torus and a projective rational homogeneous space, we call it a manifold of type III. We considered these kinds of manifolds in [8,10]. There always exists an extremal metric in any Kähler class.…”

Section: Guanmentioning

confidence: 99%

“…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: 99%

“…Hence, we may apply the method of counting zeros in [10,12] to this circumstance. −1 Q 1 ( ) is proportional to " Q" in [10]. Therefore, the counting of zeros of − α should be the same as counting the zeros of the derivative of " Q" to "U" there.…”

Section: −1mentioning

confidence: 99%

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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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“…Our result clarifies the third class. Then we show concretely that for the homogeneous cases, our condition (10) holds. This of course should be true, but we just use it as examples.…”

Section: Introductionmentioning

confidence: 81%

Section: Guanmentioning

confidence: 98%

Section: Guanmentioning

confidence: 99%

Section: −1mentioning

confidence: 99%

Section: −1mentioning

confidence: 99%

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Affine compact almost-homogeneous manifolds of cohomogeneity one

Guan

2009

Open Mathematics

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Constant Scalar Curvature Sasaki Metrics and Projective Bundles

Boyer

Tønnesen-Friedman

2023

Springer Proceedings in Mathematics &Amp; Statistics

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Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability

et al. 2008

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This paper concerns the explicit construction of extremal Kähler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely independent of that theory.We obtain a characterization, on a large family of projective bundles, of those 'admissible' Kähler classes (i.e., the ones compatible with the bundle structure in a way we make precise) which contain an extremal Kähler metric. In many cases, such as on geometrically ruled surfaces, every Kähler class is admissible. In particular, our results complete the classification of extremal Kähler metrics on geometrically ruled surfaces, answering several long-standing questions.We also find that our characterization agrees with a notion of K-stability for admissible Kähler classes. Our examples and nonexistence results therefore provide a fertile testing ground for the rapidly developing theory of stability for projective varieties, and we discuss some of the ramifications. In particular we obtain examples of projective varieties which are destabilized by a non-algebraic degeneration.Ψ * y = y c , Ψ * t = t, and hence Ψ * J = J c .As J c and J are integrable complex structures, Ψ extends to a U (1)-equivariant diffeomorphism of M leaving fixed any point on e 0 ∪e ∞ (since it is fibre preserving).Putω := Ψ * ω. Thenω is a Kähler form on (M, J c ) which (we claim) belongs to the same cohomology class Ω as ω. Indeed, on M 0 we havẽdd c Jc h(y c ) = Ψ * dd c J h(y) =ω − a ω a /x a , so the following implies the claim. Lemma 3. The function h(y c ) − h c (y c ) is smooth on M . Centre-ville,

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Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends

Cited by 22 publications

References 6 publications

Affine compact almost-homogeneous manifolds of cohomogeneity one

Affine compact almost-homogeneous manifolds of cohomogeneity one

Constant Scalar Curvature Sasaki Metrics and Projective Bundles

Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability

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