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.

Ambitoric geometry I: Einstein metrics and extremal ambikähler structures

2016 ◽  
Vol 2016 (721) ◽  
Author(s):  
Vestislav Apostolov ◽  
David M. J. Calderbank ◽  
Paul Gauduchon
Keyword(s):  
Maxwell Equations ◽  
Einstein Metrics ◽  
Natural Extension ◽  
Torus Action ◽  
Local Geometry ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Bach Tensor ◽  
Extremal Kähler Metrics

AbstractWe present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kähler metrics which are toric with respect to a common 2-torus action. In the generic case, these “ambitoric” structures have an intriguing local geometry depending on a quadratic polynomialWe use this description to classify 4-dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors.These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures,

scholarly journals Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

2019 ◽  
Vol 2019 (751) ◽  
pp. 27-89 ◽  
Author(s):  
Robert J. Berman ◽  
Sebastien Boucksom ◽  
Philippe Eyssidieux ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Ricci Flow ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Einstein Metrics ◽  
Discrete Version ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics ◽  
Special Case

AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

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

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Daniel Guan
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Symmetric Spaces ◽  
Vector Fields ◽  
Constant Scalar Curvature ◽  
Explicit Construction ◽  
Kähler Metrics ◽  
Cohomogeneity One ◽  
Kahler Metrics ◽  
Kähler Einstein Metrics

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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