Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds

Abstract

Suppose $(X,J,\omega )$ is a Fano manifold and $t\to r_t$ is a diverging Kähler-Ricci trajectory. We construct a bounded geodesic ray $t\to u_t$ weakly asymptotic to $t\to r_t$, along which Ding's $\mathcal{F}$-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fields this proves the equivalence between geodesic stability of the $\mathcal{F}$-functional and existence of Kähler-Einstein metrics. We also explore applications of our construction to Tian's $\alpha$-invariant.

v2. Improved presentation. References added

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Cited by in corpus (2)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Tian's properness conjectures: an introduction to Kahler geometry