Metric Dependence and Asymptotic Minimization of the Expected Number of Critical Points of Random Holomorphic Sections

TL;DR

This paper proves a conjecture about how the expected number of critical points of random holomorphic sections depends on the metric, showing that the Calabi extremal metric minimizes this number asymptotically across all dimensions.

Contribution

It confirms the metric dependence of the expected critical points and establishes the asymptotic minimization by Calabi extremal metrics, extending previous conjectures.

Findings

The first non-topological term in the asymptotic expansion involves the Calabi functional.

The constant (m) is positive in all dimensions, indicating non-topological behavior.

Calabi extremal metrics asymptotically minimize the expected number of critical points.

Abstract

We prove the main conjecture from [M. R. Douglas, B. Shiffman and S. Zelditch, Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the metric dependence and asymptotic minimization of the expected number \mathcal{N}^{crit}_{N,h} of critical points of random holomorphic sections of the Nth tensor power of a positive line bundle. The first non-topological term in the asymptotic expansion of \mathcal{N}^{crit}_{N,h} is the the Calabi functional multiplied by the constant \be_2(m) which depends only on the dimension of the manifold. We prove that \be_2(m) is strictly positive in all dimensions, showing that the expansion is non-topological for all m, and that the Calabi extremal metric, when it exists, asymptotically minimizes \mathcal{N}^{crit}_{N,h}.