scholarly journals Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

1991 ◽  
Vol 136 (2) ◽  
pp. 309-326 ◽  
Author(s):  
Henrik Pedersen ◽  
Y. Sun Poon
Keyword(s):  
Scalar Curvature ◽  
Einstein Metrics ◽  
Constant Scalar Curvature ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Kähler Einstein Metrics

scholarly journals Generalizations of Kähler-Ricci solitons on projective bundles

2011 ◽  
Vol 108 (2) ◽  
pp. 161 ◽  
Author(s):  
Gideon Maschler ◽  
Christina W. Tønnesen-Friedman
Keyword(s):  
Scalar Curvature ◽  
Existence Theorem ◽  
Einstein Metrics ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Projective Bundles ◽  
Affine Combination ◽  
Constant Scalar Curvature Metrics

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and Tønnesen-Friedman), arising from a base with a local Kähler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein Kähler metrics (as defined by D. Guan) in all "sufficiently small" admissible Kähler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some Kähler classes while not in others. We also prove an analogous existence theorem for an additional metric type, defined by the requirement that the scalar curvature is an affine combination of a Killing potential and its Laplacian.

scholarly journals Geometry and topology of the space of Kähler metrics on singular varieties

2018 ◽  
Vol 154 (8) ◽  
pp. 1593-1632 ◽  
Author(s):  
Eleonora Di Nezza ◽  
Vincent Guedj
Keyword(s):  
Einstein Metrics ◽  
Normal Space ◽  
Fano Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Metric Properties ◽  
Singular Varieties ◽  
Underlying Space ◽  
And Topology ◽  
Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

scholarly journals Vector Field Energies and Critical Metrics on Kähler Manifolds

2001 ◽  
Vol 162 ◽  
pp. 41-63 ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Bilinear Pairing ◽  
Fano Varieties ◽  
Holomorphic Vector Field ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kähler Einstein Metrics ◽  
Compact Kähler Manifold

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals whose critical points are extremal Kähler metrics or “Kähler-Einstein metrics” in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of , the uniqueness of “Kähler-Einstein metrics” on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

scholarly journals Remarks on extremal Kähler metrics on ruled manifolds

1992 ◽  
Vol 126 ◽  
pp. 89-101 ◽  
Author(s):  
Akira Fujiki
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Metric ◽  
Kähler Metrics ◽  
Algebraic Subgroup ◽  
Kähler Metric ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kahler Metric ◽  
Compact Kähler Manifold

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.

scholarly journals KÄHLER GEOMETRY OF TORIC VARIETIES AND EXTREMAL METRICS

1998 ◽  
Vol 09 (06) ◽  
pp. 641-651 ◽  
Author(s):  
MIGUEL ABREU
Keyword(s):  
Scalar Curvature ◽  
Convex Polytopes ◽  
Constant Scalar Curvature ◽  
Affine Function ◽  
Combinatorial Identity ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Total Integral ◽  
First Chern Class ◽  
Moment Polytope

A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn. Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kähler metrics on X, using only data on Δ. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler–Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on Δ ⊂ ℝn. A construction, due to Calabi, of a 1-parameter family of extremal Kähler metrics of non-constant scalar curvature on [Formula: see text] is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known relation between the total integral of the scalar curvature of a Kähler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kähler class.

scholarly journals An ℓ-th root of a test configuration of exponent ℓ

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Scalar Curvature ◽  
Moduli Space ◽  
Constant Scalar Curvature ◽  
Algebraic Manifold ◽  
Positive Real ◽  
Kähler Metrics ◽  
Test Configuration ◽  
Kahler Metrics ◽  
Integral Divisor

AbstractLet (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : ^X → X and η : ^X → Y, both isomorphic on^X \^X 0, such thatwhereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor on^X such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way thatwhere C1 and C2 are positive real constants independent of the choice of μ and ℓ. This plays an important role in our forthcoming papers on the existence of constant scalar curvature Kähler metrics (cf. [6]) and also on the compactified moduli space of test configurations (cf. [5],[7]).

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