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A variational approach to the Yau-Tian-Donaldson conjecture

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arxiv 1509.04561 v3 pith:HHNXHUNP submitted 2015-09-15 math.DG math.AG

Robert Berman , Sébastien Boucksom , Mattias Jonsson This is my paper · ORCID
classification math.DGmath.AG
keywords approachconjecturetheorytwistedvariationalyau-tian-donaldsonahler-einsteinalgebro-geometric
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We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted K\"ahler-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 the continuity method or 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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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The metric geometry of singularity types

    math.DG 2019-09 accept novelty 8.0 of 10

    A new metric d_S on singularity types of θ-psh potentials is defined; on positive-mass classes it is complete, and it governs convergence of solutions and multiplier ideal sheaves.

  2. The K\"ahler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations

    math.DG 2019-09 conditional novelty 7.0 of 10

    For any Fano manifold, the Donaldson-Futaki invariant of its optimal degeneration is bounded below by -(1-R(X))/R(X) nV, where R(X) is the greatest Ricci lower bound.

  3. Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons

    math.DG 2019-08 conditional novelty 7.0 of 10

    Uniform relative Ding stability of a Fano manifold implies the necessary numerical condition ϑ(M) < 1 for the existence of Mabuchi solitons.

  4. Existence of Kahler-Ricci solitons on smoothable Q-Fano varities

    math.DG 2019-08 conditional novelty 6.0 of 10

    K-stable smoothable Q-Fano varieties admit Kähler-Ricci solitons, extending the Yau-Tian-Donaldson correspondence for solitons from the smooth case to the singular smoothable case.

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