Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gibbs polystability of Fano manifolds, stability thresholds and symmetry breaking

Rolf Andreasson, Robert J. Berman, Ludvig Svensson

TL;DR

This work extends probabilistic constructions of Kähler–Einstein metrics from Burn–fstyle setups with trivial automorphism groups to log Fano manifolds with nontrivial Aut_0 by explicitly breaking symmetry. It introduces Gibbs polystability, a notion combining GIT semistability with reduced analytic thresholds, and relates it to K-polystability via a proposed Large Deviation Principle that matches algebraic and analytic stability thresholds. The authors develop both algebraic and analytic frameworks: on curves they prove Gibbs polystability and compute explicit microscopic thresholds, and they formulate a thermodynamic picture connecting Mabuchi energy, moment maps, and convex functionals. A key analytic achievement is a sharp constrained logarithmic Hardy–Littlewood–Sobolev inequality on S^2 under a moment constraint, together with a duality framework and a Liouville-type PDE characterization of minimizers. The paper also lays out a program toward asymptotics of K-reduced partition functions, relates to arithmetic Mabuchi functionals, and discusses spontaneous symmetry breaking and its consequences, with companion papers to address Onsager vortex models and AdS/CFT contexts.

Abstract

We extend the probabilistic approach for constructing Kahler-Einstein metrics on log Fano manifolds - involving random point processes - to the case of non-discrete automorphism groups, by explicitly breaking the symmetry. This yields a new algebraic notion of Gibbs polystability, conjecturally equivalent to K-polystability. The definition involves a limit of log canonical thresholds on the GIT semistable locus of the N-fold products of the Fano manifold that we conjecture coincides with an analytic reduced stability threshold encoding the coercivity of the K-energy functional modulo automorphisms. These conjectures follow from an overarching conjectural Large Deviation Principle for the limit when N tends to infinity. We prove several of our conjectures for log Fano curves. By imposing a moment constraint, we derive a strengthened form of the sharp logarithmic Hardy-Littlewood-Sobolev inequality on the two-sphere, that implies the sharp form of Aubin's refinement of the Moser-Trudinger inequality. In companion papers we will present applications to Onsager's point vortex model on the two-sphere and the AdS/CFT correspondence.

Gibbs polystability of Fano manifolds, stability thresholds and symmetry breaking

TL;DR

This work extends probabilistic constructions of Kähler–Einstein metrics from Burn–fstyle setups with trivial automorphism groups to log Fano manifolds with nontrivial Aut_0 by explicitly breaking symmetry. It introduces Gibbs polystability, a notion combining GIT semistability with reduced analytic thresholds, and relates it to K-polystability via a proposed Large Deviation Principle that matches algebraic and analytic stability thresholds. The authors develop both algebraic and analytic frameworks: on curves they prove Gibbs polystability and compute explicit microscopic thresholds, and they formulate a thermodynamic picture connecting Mabuchi energy, moment maps, and convex functionals. A key analytic achievement is a sharp constrained logarithmic Hardy–Littlewood–Sobolev inequality on S^2 under a moment constraint, together with a duality framework and a Liouville-type PDE characterization of minimizers. The paper also lays out a program toward asymptotics of K-reduced partition functions, relates to arithmetic Mabuchi functionals, and discusses spontaneous symmetry breaking and its consequences, with companion papers to address Onsager vortex models and AdS/CFT contexts.

Abstract

We extend the probabilistic approach for constructing Kahler-Einstein metrics on log Fano manifolds - involving random point processes - to the case of non-discrete automorphism groups, by explicitly breaking the symmetry. This yields a new algebraic notion of Gibbs polystability, conjecturally equivalent to K-polystability. The definition involves a limit of log canonical thresholds on the GIT semistable locus of the N-fold products of the Fano manifold that we conjecture coincides with an analytic reduced stability threshold encoding the coercivity of the K-energy functional modulo automorphisms. These conjectures follow from an overarching conjectural Large Deviation Principle for the limit when N tends to infinity. We prove several of our conjectures for log Fano curves. By imposing a moment constraint, we derive a strengthened form of the sharp logarithmic Hardy-Littlewood-Sobolev inequality on the two-sphere, that implies the sharp form of Aubin's refinement of the Moser-Trudinger inequality. In companion papers we will present applications to Onsager's point vortex model on the two-sphere and the AdS/CFT correspondence.

Paper Structure

This paper contains 70 sections, 50 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Recap of the case when $\text{Aut}_{0}(X)$ is trivial
  3. Gibbs polystability and microscopic stability thresholds
  4. Explicit symmetry breaking and the emergence of a unique Kähler--Einstein metric
  5. Outline of the proof of Theorem \ref{['thm:conv towards inf for n one intro']} and connections to arithmetic geometry
  6. Outlook on valuative aspects
  7. Acknowledgements
  8. Organization of the paper
  9. Gibbs polystability and microscopic stability thresholds of log Fano manifolds
  10. Setup
  11. Microscopic stability thresholds and Gibbs polystability
  12. Preliminaries
  13. Log canonical thresholds
  14. Semistability in GIT
  15. The Fulton--MacPherson compactification of configuration spaces
  16. ...and 55 more sections

Key Result

Theorem 1.3

The previous two conjectures hold for any log Fano curve $(X,\Delta).$ More precisely, and when the group $\mathcal{G}$ is non-trivial $(X,\Delta)$ is Gibbs polystable iff $\Delta$ is of the form $\Delta_{w}$ for some $w\in[0,1[.$ Moreover, for $N>3$, For $N=2,3$, $\gamma^{(N)}(X,0)^{\mathcal{G}}=\infty$, and $\gamma^{(N)}(X,\Delta_{w})^{\mathcal{G}}=\frac{2(1-1/N)}{V}$. Additionally,

Theorems & Definitions (111)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 101 more