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Number variance of random zeros on complex manifolds, II: smooth statistics

We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic formulas show that the variances for these smooth statistics have the growth $N^{m-2}$. We also prove analogues for the integrals of smooth test forms over the subvarieties defined by $k<m$ random polynomials. Such linear statistics of random zero sets are smooth analogues of the random variables given by counting the number of zeros in an open set, which we proved elsewhere to have variances of order $N^{m-1/2}$. We use the variance asymptotics and off-diagonal estimates of Szego kernels to extend an asymptotic normality result of Sodin-Tsirelson to the case of smooth linear statistics for zero sets of codimension one in any dimension $m$.