A variational approach to the Yau–Tian–Donaldson conjecture

Berman, Robert; Boucksom, Sébastien; Jonsson, Mattias

doi:10.1090/jams/964

A variational approach to the Yau–Tian–Donaldson conjecture
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by Robert J. Berman, Sébastien Boucksom and Mattias Jonsson

J. Amer. Math. Soc. 34 (2021), 605-652

DOI: https://doi.org/10.1090/jams/964

Published electronically: April 1, 2021

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Abstract:

We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.

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