Geodesic rays and Kähler–Ricci trajectories on Fano manifolds

Darvas, Tamás; He, Weiyong

doi:10.1090/tran/6878

Geodesic rays and Kähler–Ricci trajectories on Fano manifolds
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by Tamás Darvas and Weiyong He PDF

Trans. Amer. Math. Soc. 369 (2017), 5069-5085 Request permission

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 the 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.

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