[Paper Review] Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n

Robert J. Berman|ArXiv.org|Feb 13, 2007

Geometry and complex manifolds1 references18 citations

TL;DR

This paper establishes the asymptotic behavior of Bergman kernels for weighted polynomials in ℂⁿ under a 𝒞¹,¹ weight function 𝜙, showing that the rescaled Bergman function converges weakly to a weighted equilibrium measure supported on a compact set D. The key result is that all three natural measures—Bergman function, Bergman kernel density, and Monge-Ampère measure of the equilibrium potential—coincide with the measure 1_D (ddᶜ𝜙)^n / n! in the large k limit, extending known results from n=1 to higher dimensions and linking to random matrix theory and random polynomials.

ABSTRACT

Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.

Motivation & Objective

• To analyze the large k asymptotics of Bergman kernels for weighted polynomials in ℂⁿ with 𝒞¹,¹ weight functions.
• To identify the limiting measure of the rescaled Bergman function k⁻ⁿBₖ(𝑧)𝜔ₙ as k→∞.
• To unify three natural measures—Bergman function, Bergman kernel density, and Monge-Ampère measure—under the same equilibrium measure on a compact set D.
• To extend known results from n=1 to higher dimensions, particularly for random matrix eigenvalues and random polynomial zero distributions.
• To establish local asymptotic expansions of the Bergman kernel in powers of k, matching the Tian-Zelditch-Catlin expansion in the smooth, positive curvature case.

Proposed method

• Use of the weighted Hilbert space ℋₖ of polynomials of total degree ≤k with norm ||𝑓ₖ||²ₖᵠ := ∫|𝑓ₖ(𝑧)|²e⁻ᵏᶲ d𝜔ₙ.
• Definition of the Bergman kernel Kₖ(𝑧,𝑤) and Bergman function Bₖ(𝑧) := Kₖ(𝑧,𝑧)e⁻ᵏᶲ.
• Introduction of the weighted equilibrium potential 𝜙ₑ as the upper envelope of subharmonic functions majorized by 𝜙.
• Application of the Monge-Ampère measure (ddᶜ𝜙ₑ)ⁿ/n! and its identification with the limiting measure 1_D (ddᶜ𝜙)^n / n!.
• Use of the complex torus action 𝑧 ↦ eʰ𝑧 to generalize regularity and comparison results from ℝⁿ to ℂ*ⁿ.
• Establishment of L¹ convergence of k⁻ⁿBₖ to 1_D det(ddᶜ𝜙) on ℂ*ⁿ, using volume estimates and lattice point counting for Newton polytopes.

Experimental results

Research questions

• RQ1 How does the Bergman function Bₖ(𝑧) behave in the large k limit for weighted polynomials in ℂⁿ with a 𝒞¹,¹ weight function?
• RQ2 What is the weak limit of the measure k⁻ⁿBₖ(𝑧)𝜔ₙ as k→∞, and how does it relate to the curvature of the weight function?
• RQ3 Can the convergence of the Bergman kernel and related measures be extended from n=1 to higher dimensions, and under what regularity assumptions?
• RQ4 To what extent do the eigenvalue distributions of random normal matrices and the zero sets of random polynomials converge to the same equilibrium measure in ℂⁿ?
• RQ5 Under what conditions does the Bergman kernel admit a complete local asymptotic expansion in powers of k, and how does it relate to the Tian-Zelditch-Catlin expansion?

Key findings

• The rescaled Bergman function k⁻ⁿBₖ(𝑧) converges weakly to the measure 1_D (ddᶜ𝜙)^n / n! on ℂⁿ as k→∞, where D is a compact set defined by the weight function.
• The three measures—k⁻ⁿBₖ(𝑧)𝜔ₙ, (ddᶜ(k⁻¹ln Kₖ(𝑧,𝑧)))ⁿ/n!, and (ddᶜ𝜙ₑ)^n/n!—all converge weakly to the same equilibrium measure 1_D (ddᶜ𝜙)^n / n!.
• For smooth points in the interior of D with ddᶜ𝜙 > 0, the Bergman kernel Kₖ(𝑧,𝑤) admits a complete local asymptotic expansion in powers of k, matching the Tian-Zelditch-Catlin expansion.
• The global weak limit of k⁻ⁿ|Kₖ(𝑧,𝑤)|²e⁻ᵏᶲ(𝑧)e⁻ᵏᶲ(𝑤)𝜔ₙ(𝑧)∧𝜔ₙ(𝑤) is 1_D Δ ∧ (ddᶜ𝜙)^n / n! on ℂⁿ×ℂⁿ, where Δ is the diagonal current.
• For weighted polynomials associated to a Newton polytope kΔ, the rescaled Bergman function converges in L¹ to 1_{D_Δ ∩ ℂ*ⁿ} det(ddᶜ𝜙), with weak convergence to the equilibrium measure (ddᶜ𝜙_Δ,ₑ)^n / n!.
• The volume of the interior of the Newton polytope Δ provides a lower bound for dim ℋₖΔ, with dim ℋₖΔ ≥ Vol(Δ̇)kⁿ + o(kⁿ), and equality holds in the asymptotic limit.