K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics

Abstract

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functional along weak geodesic rays in the space of all bounded positively curved metrics on the anti-canonical line bundle of X and also gives a new proof in the non-singular case. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einstein metrics with edge-cone singularities. Furthermore, applications to geodesic stability, bounds on the Ricci potential and Perelman's entropy functional on K-unstable Fano manifolds are given.

v1: 20 pages, v2: 32 pages. Substantial revision. The mean new features are: an explicit formula for the Donaldson-Futaki invariant of a general (and not only special) test configuration in terms of the slope of the Ding functional. v3: 41 pages. Exposition improved and application to geodesic stability added - this is the final version to appear in Inventiones

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• A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties
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• Kähler-Einstein metrics and volume minimization
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• $\cF$-functional and geodesic stability
• Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds
• Minimizing normalized volumes of valuations
• Compact moduli space of Kahler-Einstein Fano varieties
• Orthogonal projection of a test configuration to vector fields
• On the limit of spectral measures associated to a test configuration
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• Projective embedding of log Riemann surfaces and K-stability
• Examples of K-unstable Fano manifolds with the Picard number one
• On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics
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• Cosmology: macro and micro
• On log K-stability for asymptotically log Fano varieties