  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.
