Compactness of Kähler metrics with bounds on Ricci curvature and $${\mathcal {I}}$$ functional

2019 ◽  
Vol 58 (4) ◽  
Author(s):  
Xiuxiong Chen ◽  
Tamás Darvas ◽  
Weiyong He
Keyword(s):  
Ricci Curvature ◽  
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 Adiabatic limits of Ricci-flat Kähler metrics

2010 ◽  
Vol 84 (2) ◽  
pp. 427-453 ◽  
Author(s):  
Valentino Tosatti
Keyword(s):  
Kähler Metrics ◽  
Kahler Metrics ◽  
Adiabatic Limits ◽  
Ricci Flat
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