  Linear statistics of random zero sets are integrals of smooth differential
forms over the zero set and as such are smooth analogues of the volume of the
random zero set inside a fixed domain. We derive an asymptotic expansion for
the variance of linear statistics of the zero divisors of random holomorphic
sections of powers of a positive line bundle on a compact K\"ahler manifold.
This expansion sharpens the leading-order asymptotics (in the codimension one
case) given by Shiffman--Zelditch in 2010.
