Gibbs polystability of Fano manifolds, stability thresholds and symmetry breaking

Abstract

We extend the probabilistic approach for constructing Kähler--Einstein metrics on log Fano manifolds $(X,Δ)$ - involving random point processes - to the case of non-discrete automorphism groups, by breaking the symmetry using a moment map constraint. In particular, an algebraic notion of Gibbs polystability is introduced, ensuring that the corresponding point processes on $X$ are well-defined. We conjecture that the Gibbs polystability of $(X,Δ)$ is equivalent to the existence of a Kähler--Einstein metric and that the unique such metric with vanishing moment emerges when sampling a large number of $N$ points on $X$. The definition of Gibbs polystability is formulated in terms of a new algebro-geometric invariant, defined as a limit of certain log canonical thresholds on the GIT semistable locus of the $N$-fold products $X^N$. We conjecture that it coincides with a previously studied analytic polystability threshold, encoding the coercivity of the K-energy functional modulo automorphisms. These conjectures follow from an overarching conjectural Large Deviation Principle for the large $N$-limit. The conjectured equivalence between Gibbs polystability and K-polystability is established when $(X,Δ)$ is a log Fano curve. For Fano manifolds of any dimension, we establish an effective one-sided version of the conjecture, which yields an effective criterion for the existence of a Kähler--Einstein metric and, more generally, an effective algebraic lower bound on the analytic polystability threshold. In companion papers, applications will be given to optimal stability results for the logarithmic HLS inequality on the two-sphere, to Onsager's point vortex model on the two-sphere, and to the AdS/CFT correspondence.