  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 $m$.
