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

Abstract. We prove the main conjecture from [DSZ2] concerning the metric dependence and asymptotic minimization of the expected number N crit N,h of critical points of random holomorphic sections of the Nth tensor power of a positive line bundle. The first nontopological term in the asymptotic expansion of N crit N,h is the the Calabi functional multiplied by the constant β2(m) which depends only on the dimension of the manifold. We prove that β2(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 N crit N,h.