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

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}.