  The main results of this article are asymptotic formulas for the variance of
the number of zeros of a Gaussian random polynomial of degree $N$ in an open
set $U \subset C$ as the degree $N \to \infty$, and more generally for the
zeros of random holomorphic sections of high powers of any positive line bundle
over any Riemann surface. The formulas were conjectured in special cases by
Forrester and Honner. In higher dimensions, we give similar formulas for the
variance of the volume inside a domain $U$ of the zero hypersurface of a random
holomorphic section of a high power of a positive line bundle over any compact
K\"ahler manifold. These results generalize the variance asymptotics of Sodin
and Tsirelson for special model ensembles of chaotic analytic functions in one
variable to any ample line bundle and Riemann surface. We also combine our
methods with those of Sodin-Tsirelson to generalize their asymptotic normality
results for smoothed number statistics.
