  We comprehensively analyse the cosmology dependence of counts-in-cell
statistics. We focus on the shape of the one-point probability distribution
function (PDF) of the matter density field at mildly nonlinear scales. Based on
large-deviation statistics, we parametrise the cosmology dependence of the
matter PDF in terms of the linear power spectrum, the growth factor, the
spherical collapse dynamics, and the nonlinear variance. We extend our
formalism to include massive neutrinos, finding that the total matter PDF is
highly sensitive to the total neutrino mass $M_\nu$ and can disentangle it from
the clustering amplitude $\sigma_8$. Using more than a million PDFs extracted
from the Quijote simulations, we determine the response of the matter PDF to
changing parameters in the $\nu\Lambda$CDM model and successfully
cross-validate the theoretical model and the simulation measurements. We
present the first $\nu\Lambda$CDM Fisher forecast for the matter PDF at
multiple scales and redshifts, and its combination with the matter power
spectrum. We establish that the matter PDF and the matter power spectrum are
highly complementary at mildly nonlinear scales. The matter PDF is particularly
powerful for constraining the matter density $\Omega_m$, clustering amplitude
$\sigma_8$ and the total neutrino mass $M_\nu$. Adding the mildly nonlinear
matter PDF to the mildly nonlinear matter power spectrum improves constraints
on $\Omega_m$ by a factor of 5 and $\sigma_8$ by a factor of 2 when considering
the three lowest redshifts. In our joint analysis of the matter PDF and matter
power spectrum at three redshifts, the total neutrino mass is constrained to
better than 0.01 eV with a total volume of 6 (Gpc/h)$^3$. We discuss how
density-split statistics can be used to translate those encouraging results for
the matter PDF into realistic observables in galaxy surveys.
