  We prove that the expected value and median of the supremum of $L^2$
normalized random holomorphic fields of degree $n$ on $m$-dimensional K\"ahler
manifolds are asymptotically of order $\sqrt{m\log n}$. This improves the prior
result of Shiffman-Zelditch (arXiv:math/0303335) that the upper bound of the
media is of order $\sqrt{\log n}$
  The estimates are based on the entropy methods of Dudley and Sudakov combined
with a precise analysis of the relevant pseudo-metric and its covering numbers,
which can be precisely evaluated using off-diagonal asymptotics of Bergman
kernels. Recent work of the authors on the value distribution of these fields
are also used to get precise constants.
