The Moduli Space of Extremal Compact Kähler Manifolds and Generalized Well-Petersson Metrics

Fujiki, Akira; Schumacher, Georg

doi:10.2977/prims/1195171664

Cited by 128 publications

(163 citation statements)

References 35 publications

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“…Passing to the universal covering X = C X P we see that Aut o (^\ g) is covered by a product group of the form C X Aut(P, gp), where £p is some Kahler metric on P. Especially, the induced Kahler metric g on X is invariant by the C-action on the first factor. We now consider the Kunneth decomposition as (8) for the Kahler form co -(Dg on C X P; for example a>n = <x>n(0 (resp. o> 2 2 -£^22(0) is a C°°f amily of (l,l)-forms on C (resp.…”

Section: ) If L(x) Is Unipotent and Is Of Positive Dimension Then X mentioning

confidence: 99%

Remarks on extremal Kähler metrics on ruled manifolds

Fujiki

1992

Nagoya Mathematical Journal

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Let X be a compact Kähler manifold and γ Kähler class. For a Kàhler metric g on X we denote by Rg the scalar curvature on X According to Calabi [3][4], consider the functional defined on the set of all the Kähler metrics g whose Kähler forms belong to γ, where dvg is the volume form associated to g. Such a Kähler metric is called extremal if it gives a critical point of Ф. In particular, if Rg is constant, g is extremal. The converse is also true if dim L(X) = 0, where L(X) is the maximal connected linear algebraic subgroup of AutoX (cf. [5]). Note also that any Kähler-Einstein metric is of constant scalar curvature.

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Section: ) If L(x) Is Unipotent and Is Of Positive Dimension Then X mentioning

confidence: 99%

Remarks on extremal Kähler metrics on ruled manifolds

Fujiki

1992

Nagoya Mathematical Journal

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“…This result of [4] was extended to (smooth) Kähler fibrations over singular base spaces in [8,Sec. 12].…”

Section: The Curvaturementioning

confidence: 90%

Deligne pairing and Quillen metric

Biswas

Schumacher

2014

Int. J. Math.

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Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over C. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing L, . . . , L → S with the determinant line bundle Det((L − O X ) ⊗(n+1) ⊗ L ⊗m ) (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen-Mumford expansion, J. Differential Geom. 78 (2008) 475-496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles Det((L − O X ) ⊗(n+1) ⊗ L ⊗m ) and L, . . . , L carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between L, . . . , L → S and Det((L−O X ) ⊗(n+1) ⊗L ⊗m ) is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91-96] between a Deligne paring and a certain determinant line bundle.

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“…In the Kähler setting, Fujiki-Schumacher [9] and Lebrun-Simanca [14] showed, in the abscence of holomorphic vector fields, that the existence of extremal Kähler metrics is an open condition. Moreover, ApostolovCalderbank-Gauduchon-Friedman [1] proved the openess by fixing a maximal torus in the reduced automorphism group of the complex manifold (M, J).…”

Section: Introductionmentioning

confidence: 99%

Stability under deformations of Hermite-Einstein almost Kähler metrics

Lejmi¹

2014

Ann. inst. Fourier

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The Moduli Space of Extremal Compact Kähler Manifolds and Generalized Well-Petersson Metrics

Cited by 128 publications

References 35 publications

Remarks on extremal Kähler metrics on ruled manifolds

Remarks on extremal Kähler metrics on ruled manifolds

Deligne pairing and Quillen metric

Stability under deformations of Hermite-Einstein almost Kähler metrics

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