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Geodesic rays and Kähler-Ricci trajectories on Fano manifolds


Authors: Tamás Darvas and Weiyong He
Journal: Trans. Amer. Math. Soc. 369 (2017), 5069-5085
MSC (2010): Primary 53C55, 32W20, 32U05
DOI: https://doi.org/10.1090/tran/6878
Published electronically: March 1, 2017
MathSciNet review: 3632560
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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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Additional Information

Tamás Darvas
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: tdarvas@math.umd.edu

Weiyong He
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: whe@uoregon.edu

DOI: https://doi.org/10.1090/tran/6878
Received by editor(s): January 9, 2015
Received by editor(s) in revised form: November 17, 2015
Published electronically: March 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society