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arXiv · 1411.0774

Geodesic Rays and K\"ahler-Ricci Trajectories on Fano Manifolds

Tamás Darvas·Weiyong He
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Abstract

Suppose $(X,J,\omega)$ is a Fano manifold and $t \to r_t$ is a diverging K\"ahler-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 K\"ahler-Einstein metrics. We also explore applications of our construction to Tian's $\alpha$-invariant.

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Tamás Darvas, Weiyong He. 2014-11-04. Geodesic Rays and K\"ahler-Ricci Trajectories on Fano Manifolds. https://arxiv.org/abs/1411.0774

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