Number Variance of Random Zeros on Complex Manifolds

Abstract.

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 $U \subset {\mathbb{C}}^m$ with smooth boundary is asymptotic to $N^{{m-1}/2} \nu_{mm} {\rm Vol}(\partial U)$, 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 ${\mathbb{C}}^{m}$. 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 Kähler manifold.