  We show that the variance of the number of simultaneous zeros of $m$ i.i.d.
Gaussian random polynomials of degree $N$ in an open set $U \subset C^m$ with
smooth boundary is asymptotic to $N^{m-1/2} \nu_{mm} Vol(\partial U)$, where
$\nu_{mm}$ is a universal constant depending only on the dimension $m$. We also
give formulas for the variance of the volume of the set of simultaneous zeros
in $U$ of $k<m$ random degree-$N$ polynomials on $C^m$. Our results hold more
generally for the simultaneous zeros of random holomorphic sections of the
$N$-th power of any positive line bundle over any $m$-dimensional compact
K\"ahler manifold.
