  We construct a splitting of the cohomology of configuration spaces of points
on a smooth proper variety with a multiplicative Chow--K\"unneth decomposition.
Applied to hyperelliptic curves, this shows that the hyperelliptic Torelli
group acts trivially on the rational cohomology of ordered configuration spaces
of points. Moreover, if $H_{g,n}$ denotes the moduli space of $n$-pointed
hyperelliptic curves, the Leray spectral sequence for the forgetful map
$H_{g,n} \to H_g$ degenerates immediately, in sharp contrast to the forgetful
map from $M_{g,n}$ to $M_g$. This allows for new detailed calculations of the
cohomology of $M_{2,n}$ for $n \leq 5$, and the stable cohomology of $H_{g,n}$
for $n \leq 5$. We also give a detailed study of the cohomology of symplectic
local systems on $M_2$.
