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

Abstract:

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 $\mathcal {N}^{\operatorname {crit}}_{N,h}$.