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A variational approach to the Yau–Tian–Donaldson conjecture


Authors: Robert J. Berman, Sébastien Boucksom and Mattias Jonsson
Journal: J. Amer. Math. Soc. 34 (2021), 605-652
MSC (2020): Primary 32Q20, 32Q26
DOI: https://doi.org/10.1090/jams/964
Published electronically: April 1, 2021
MathSciNet review: 4334189
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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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Additional Information

Robert J. Berman
Affiliation: Mathematical Sciences, Chalmers University of Technology; and University of Gothenburg, SE-412 96 Göteborg, Sweden
MR Author ID: 743613
Email: robertb@chalmers.se

Sébastien Boucksom
Affiliation: CNRS-CMLS, École Polytechnique, F-91128 Palaiseau Cedex, France
MR Author ID: 688226
Email: sebastien.boucksom@polytechnique.edu

Mattias Jonsson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
MR Author ID: 631360
Email: mattiasj@umich.edu

Received by editor(s): January 24, 2019
Received by editor(s) in revised form: June 2, 2020
Published electronically: April 1, 2021
Additional Notes: The first author was partially supported by the Swedish Research Council, the European Research Council, the Knut and Alice Wallenberg foundation, and the Göran Gustafsson foundation.
The second author was partially supported by the ANR projects GRACK, MACK and POSITIVE.
The third author was partially supported by NSF grants DMS-1600011 and DMS-1900025, the Knut and Alice Wallenberg foundation, and the United States—Israel Binational Science Foundation.
Article copyright: © Copyright 2021 American Mathematical Society