Number variance of random zeros on complex manifolds

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.