International audienc

It is shown that any, possibly singular, Fano variety X admitting a Kähler-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 is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X . 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 Kähler-Einstein metrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's λ-entropy functional on K -unstable Fano manifolds are also given.

Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e −φ on L, we define the energy at equilibrium of (K, φ) as the Monge-Ampère energy of the extremal psh weight associated to (K, φ). We prove the differentiability of the energy at equilibrium with respect to φ, and we show that this energy describes the asymptotic behaviour as k → ∞ of the volume of the sup-norm unit ball induced by (K, kφ) on the space of global holomorphic sections H 0 (X, kL). As a consequence of these results, we recover and extend Rumely's Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan's equidistribution theorem for algebraic points of small height to the case of a big line bundle.