[Paper Review] Weighted Pluripotential Theory Results of Berman-Boucksom

N. Levenberg|arXiv (Cornell University)|Oct 19, 2010

Geometry and complex manifolds24 references16 citations

TL;DR

This paper provides a comprehensive exposition of Berman-Boucksom's foundational results in weighted pluripotential theory, establishing the asymptotic distribution of weighted Fekete points and optimal measures to the weighted Monge-Ampère measure $\mu_{K,Q}$, and proving a weighted version of Rumely's formula linking transfinite diameter to energy integrals via the Robin function. The key contribution is a rigorous derivation of the weighted transfinite diameter formula using circled sets and logarithmic homogeneity in $\mathbb{C}^{d+1}$, with applications to strong Bergman asymptotics and Bernstein-Markov properties.

ABSTRACT

The main goal of these notes, compiled in 2008-2009, is to present a more-or-less self-contained discussion of some of the recent results and techniques of R. Berman and S. Boucksom in the setting of weighted pluripotential theory. We include some background results on pluripotential theory and weighted pluripotential theory, although many items are stated without proof (references are provided). Although slightly dated and certainly not error-free, we hope someone finds them helpful.

Motivation & Objective

• To present a self-contained, accessible exposition of Berman-Boucksom's recent advances in weighted pluripotential theory for researchers unfamiliar with the field.
• To establish the asymptotic equidistribution of weighted Fekete arrays and optimal measures toward the weighted Monge-Ampère measure $\mu_{K,Q}$.
• To prove a weighted version of Rumely's formula relating the weighted transfinite diameter $d^w(K)$ to the energy integral of the weighted extremal function $V_{K,Q}^*$.
• To develop strong Bergman asymptotics for Bernstein-Markov pairs and triples, essential for understanding extremal functions and measures in complex analysis.
• To clarify the role of weighted pluripotential theory in proving even unweighted results, demonstrating its foundational necessity.

Proposed method

• The proof of Rumely's formula uses a circled set construction $F = \{(t,z) \in \mathbb{C}^{d+1} : |t| = w(\lambda), \lambda \in K\}$, which is shown to be regular and compact.
• Logarithmic homogeneity of the relative extremal function $\rho_F$ is exploited to relate $V_{K,Q}(\lambda)$ to $\rho_F(1,\lambda)$ and $\rho_F(0,\lambda)$.
• The weighted transfinite diameter $d^w(K)$ is identified with the unweighted transfinite diameter $\delta(K^w_\rho)$ of the set $K^w_\rho = \{\lambda : \rho_{K,Q}(\lambda) \leq 0\}$, which equals $F \cap \{t=0\}$.
• The formula for $-\log \delta(F)$ in $\mathbb{C}^{d+1}$ is applied, decomposing the energy integral into terms involving $\rho_F(1,\lambda)$ and $\delta(F \cap \{t=0\})$.
• The key identity $\delta^w(K) = \delta(F)^{(d+1)/d}$ is used to link the weighted and unweighted diameters, completing the derivation of the weighted Rumely formula.
• Approximation arguments via decreasing sequences of regular compacta $K_j$ and continuous weights $w_j$ extend the result to general admissible weights and non-regular sets.

Experimental results

Research questions

• RQ1 How can the transfinite diameter of a compact set in $\mathbb{C}^d$ be expressed in terms of integrals involving the Robin function and the weighted extremal function?
• RQ2 What is the asymptotic distribution of weighted Fekete points and optimal measures in the context of weighted pluripotential theory?
• RQ3 How does the weighted Monge-Ampère measure $\mu_{K,Q}$ arise as the weak limit of weighted Fekete and optimal measures?
• RQ4 What is the precise relationship between the weighted transfinite diameter $d^w(K)$ and the energy integral $\int_K Q \, (dd^c V_{K,Q}^*)^d$?
• RQ5 How can the theory of circled sets in $\mathbb{C}^{d+1}$ be used to derive weighted versions of classical formulas in pluripotential theory?

Key findings

• The weighted transfinite diameter satisfies $\delta^w(K) = \exp\left(-\frac{1}{d} \int_K Q \, (dd^c V_{K,Q}^*)^d\right) \cdot d^w(K)$, establishing a weighted Rumely formula.
• Asymptotically, weighted Fekete arrays distribute to the weighted Monge-Ampère measure $\mu_{K,Q} = \frac{1}{(2\pi)^d} (dd^c V_{K,Q}^*)^d$.
• Optimal measures for the weighted extremal problem asymptotically distribute to $\mu_{K,Q}$, confirming a conjecture in complex approximation theory.
• Strong Bergman asymptotics hold for Bernstein-Markov pairs $(K,\mu)$ and weighted triples $(K,\mu,Q)$, generalizing classical results to the weighted setting.
• The support $S_w$ of $\mu_{K,Q}$ is compact and satisfies $V_{K,Q}^* = Q$ quasi-everywhere on $S_w$, a key structural property of weighted extremal functions.
• The identity $d^w(K) = \delta(K^w_\rho)$ holds, where $K^w_\rho = \{\lambda : \rho_{K,Q}(\lambda) \leq 0\}$, linking weighted geometry to unweighted transfinite diameter.