A variational approach to the Yau-Tian-Donaldson conjecture

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 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.

Added Appendix B on a valuative analysis of singularities of plurisubharmonic functions. Various other small changes and improvements. To appear in Journal of the AMS

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• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
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• Continuity of delta invariants and twisted Kähler--Einstein metrics
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• Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold
• Mabuchi geometry of big cohomology classes with prescribed singularities
• Relative Ding and $K$-stability of toric Fano manifolds in low dimensions
• A variational approach to the Hermitian-Einstein metrics and the Quot-scheme limit of Fubini-Study metrics
• On the optimal volume upper bound for Kähler manifolds with positive Ricci curvature (with an appendix by Yuchen Liu)
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• Openness of uniformly valuative stability on the Kähler cone of projective manifolds
• Delta invariants of smooth cubic surfaces
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• A decomposition theorem for $\mathbb{Q}$-Fano Kähler-Einstein varieties
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• About J-flow, J-balanced metrics, uniform J-stability and K-stability
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• L^1 metric geometry of big cohomology classes
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