Geometry and Topology of the space of Kähler metrics on singular varieties

Abstract

Let $Y$ be a compact Kähler normal space and $\alpha \in H^{1,1}(Y,\mathbb{R})$ a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\alpha }$ of Kähler metrics in $\alpha$ using Mabuchi geodesics. We extend several results by Calabi, Chen, 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.

36 pages. arXiv admin note: text overlap with arXiv:1401.7857

References in corpus (5)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• A variational approach to the Yau-Tian-Donaldson conjecture
• On the singularity type of full mass currents in big cohomology classes
• $C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
• Probability measures related to geodesics in the space of Kähler metrics

Cited by in corpus (7)

• Wall crossing for K-moduli spaces of plane curves
• Tits buildings and K-stability
• Kähler-Einstein metrics on families of Fano varieties
• $C^{1,1}$ regularity of geodesics of singular Kähler metrics
• Weighted cscK metrics on Kähler varieties
• Geometry and Topology of the space of plurisubharmonic functions
• L^1 metric geometry of big cohomology classes