Abstract
In the present paper and the companion paper (Berman, Kähler–Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling “temperature deformed” determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kähler–Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kähler–Einstein metrics with positive Ricci curvature is developed.
Highlights
The processes are “positive temperature deformations” of determinantal point processes and the main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper
As an application we show that the unique Kähler–Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X
In the presence of a stressenergy tensor on X it is shown that the unique Kähler metric solving Einstein’s equation on X with negative cosmological constant emerges in the many particle limit
Summary
Ric g ≥ −κ2(n − 1)g for some positive constant κ (sometimes referred to as the normalized lower bound on the Ricci curvature). Before turning to the proof we point out that it is well-known that submean inequalities with a multiplicative constant C(n) do hold in the more general singular setting of Alexandrov spaces (with a strict lower bound −κ on the sectional curvature). The only difference from the argument used in [44] is that we have taken the point p to be of distance 1 from x0 rather than distance 2δ, as used in [44] For δ small this change has the effect of improving the exponential factor from en(1+δκ) to en(δ+δκ), which is crucial as we need a constant in the Poincare inequality which has subexponential growth in n as δ → 0. (as is seen by multiplying with a suitable smooth function χ supported on B2δ such that χ = 1 on Bδ) All in all this concludes the proof of Theorem 2.1 in the case λ = 0. B2δ(x0)⊂M v2d Vg , B δ/2(x0,0)⊂M d Vg which concludes the proof of the general case (after a suitable rescaling)
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