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Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics

Robert J. Berman

TL;DR

The paper develops a probabilistic and variational program to construct (twisted) Kähler–Einstein metrics on complex varieties by coupling pluripotential theory with microcanonical energy–entropy principles. It shows that the large-$N$ limit of microcanonical measures concentrates on maximizers $\mu^{e}$, realized as normalized KE volume forms via twisted KE solutions $\omega_{\beta}$, with a Legendre dual relation between entropy $S(e)$ and free energy $F(\beta)$. It provides a thorough treatment in both low-energy and high-energy regimes (the latter on Fano varieties), including conditional convergence results, real-analyticity, and generalizations to log Fano and general compact Kähler manifolds; it also connects these variational principles to Aubin’s continuity method and Mabuchi’s K-energy. A central theme is the Hamiltonian interpretation: a mean-field limit of the $N$-particle complex energy leads to a complex Euler–Monge–Ampère equation, positioning twisted KE metrics as stationary states of a higher-dimensional Euler-type flow and linking geometric stability to statistical-mechanical ensembles. The framework unifies complex geometry, pluripotential theory, statistical mechanics, and Hamiltonian dynamics, offering new variational characterizations of KE metrics and insights into stability phenomena in Kähler geometry. The results illuminate how large-scale geometric structures emerge from microscopic Hamiltonian dynamics on point configurations, with potential implications for understanding canonical metrics in singular and non-smooth settings.

Abstract

We introduce new probabilistic and variational constructions of (twisted) Kähler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional incompressible fluids. The probabilistic construction involves microcanonical measures associated with the level sets of the pluricomplex energy, which give rise to maximum entropy principles. These, in turn, yield novel characterizations of Kähler-Einstein metrics, as well of Fano varieties admitting such metrics. Additionally, connections to Hamiltonian dynamics are uncovered, resulting in a new evolution equation, that generalizes both the 2D incompressible Euler equation and the 2D semi-geostrophic equation.

Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics

TL;DR

The paper develops a probabilistic and variational program to construct (twisted) Kähler–Einstein metrics on complex varieties by coupling pluripotential theory with microcanonical energy–entropy principles. It shows that the large- limit of microcanonical measures concentrates on maximizers , realized as normalized KE volume forms via twisted KE solutions , with a Legendre dual relation between entropy and free energy . It provides a thorough treatment in both low-energy and high-energy regimes (the latter on Fano varieties), including conditional convergence results, real-analyticity, and generalizations to log Fano and general compact Kähler manifolds; it also connects these variational principles to Aubin’s continuity method and Mabuchi’s K-energy. A central theme is the Hamiltonian interpretation: a mean-field limit of the -particle complex energy leads to a complex Euler–Monge–Ampère equation, positioning twisted KE metrics as stationary states of a higher-dimensional Euler-type flow and linking geometric stability to statistical-mechanical ensembles. The framework unifies complex geometry, pluripotential theory, statistical mechanics, and Hamiltonian dynamics, offering new variational characterizations of KE metrics and insights into stability phenomena in Kähler geometry. The results illuminate how large-scale geometric structures emerge from microscopic Hamiltonian dynamics on point configurations, with potential implications for understanding canonical metrics in singular and non-smooth settings.

Abstract

We introduce new probabilistic and variational constructions of (twisted) Kähler-Einstein metrics on complex projective algebraic varieties, drawing inspiration from Onsager's statistical mechanical model of turbulence in two-dimensional incompressible fluids. The probabilistic construction involves microcanonical measures associated with the level sets of the pluricomplex energy, which give rise to maximum entropy principles. These, in turn, yield novel characterizations of Kähler-Einstein metrics, as well of Fano varieties admitting such metrics. Additionally, connections to Hamiltonian dynamics are uncovered, resulting in a new evolution equation, that generalizes both the 2D incompressible Euler equation and the 2D semi-geostrophic equation.

Paper Structure

This paper contains 58 sections, 45 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Complex-geometric setup
  3. Energy
  4. Entropy
  5. Main results in the low energy regime $(e\leq e_{0})$
  6. Main results in the high energy regime on Fano varieties $(e>e_{0})$
  7. Relations to Hamiltonian dynamics
  8. Relations to statistical mechanics and previous results
  9. Equivalence of the microcanonical and canonical ensembles
  10. The infimum of the Mabuchi functional and Aubin's method of continuity
  11. Point vortices
  12. Acknowledgements
  13. General (abstract) results
  14. Preliminaries
  15. Concavity
  16. ...and 43 more sections

Key Result

Theorem 1.1

Given $e\in]0,e_{0}]$ and $\epsilon\in]0,\infty],$ and $\delta_{N}$ converges in probability towards the unique maximum entropy measure $\mu^{e}$ of energy $e.$ Additionally, if $e\in]0,e_{0}[,$ the convergence is exponential, i.e. for any given $\delta>0$ there exists $C>0$ such that where $d$ denotes a fixed metric on $\mathcal{P}(X),$ metrizing the weak topology.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 74 more