Expected number of critical points of random holomorphic sections over complex projective space

We study the high dimensional asymptotics of the expected number of critical points of a given Morse index of Gaussian random holomorphic sections over complex projective space. We explicitly compute the exponential growth rate of the expected number of critical points of the largest index and of diverging indices at various rates as well as the exponential growth rate for the expected number of critical points (regardless of index). We also compute the distribution of the critical values for the expected number of critical points of smallest index.
Comment: Small change to proof in page 8, added NSF in acknowledgements