scholarly journals Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

2010 ◽  
Vol 3 (3) ◽  
pp. 295-358 ◽  
Author(s):  
Jian Song ◽  
Steve Zelditch
Keyword(s):  
Toric Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Bergman Metrics

Bergman metrics and geodesics in the space of Kähler metrics on principally polarized abelian varieties

2011 ◽  
Vol 11 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Renjie Feng
Keyword(s):  
Abelian Varieties ◽  
Kähler Manifold ◽  
Invariant Metrics ◽  
Kahler Geometry ◽  
Kähler Metrics ◽  
Complete Asymptotic Expansion ◽  
Kahler Metrics ◽  
Infinite Dimensional ◽  
Kahler Manifold ◽  
Bergman Metrics

AbstractIt is well known in Kähler geometry that the infinite-dimensional symmetric space $\mathcal{H}$ of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds $\mathcal{B}_k\subset\mathcal{H}$ of Bergman metrics of height k. Then it is natural to ask whether geodesics in $\mathcal{H}$ can be approximated by Bergman geodesics in $\mathcal{B}_k$. For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

scholarly journals Kähler metrics on singular toric varieties

2008 ◽  
Vol 238 (1) ◽  
pp. 27-40 ◽  
Author(s):  
Dan Burns ◽  
Victor Guillemin ◽  
Eugene Lerman
Keyword(s):  
Toric Varieties ◽  
Kähler Metrics ◽  
Kahler Metrics

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 Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D

2011 ◽  
Vol 29 (2) ◽  
pp. 025003 ◽  
Author(s):  
L C de Andrés ◽  
M Fernández ◽  
S Ivanov ◽  
J A Santisteban ◽  
L Ugarte ◽  
...  
Keyword(s):  
Bianchi Type ◽  
Type A ◽  
Kähler Metrics ◽  
Kahler Metrics
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