Metric dependence and asymptotic minimization of the expected number of critical points of random holomorphic sections

We prove the main conjecture from Douglas, Shiffman, and Zelditch (2006) concerning the metric dependence and asymptotic minimization of the expected number $\mathcal{N}^{\operatorname{crit}}_{N,h}$ of critical points of random holomorphic sections of the $N$th tensor power of a positive line bundle. The first non-topological term in the asymptotic expansion of $\mathcal{N}^{\operatorname{crit}}_{N,h}$ is the Calabi functional multiplied by the constant $\beta_2(m)$ which depends only on the dimension of the manifold. We prove that $\beta_2(m)$ 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 $\mathcal{N}^{\operatorname{crit}}_{N,h}$.