  Although the Cold + Hot Dark Matter (CHDM) cosmology provides perhaps the
best fit of any model to all the available data at the current epoch, CHDM
produces structure at relatively low redshifts and thus could be ruled out if
there were evidence for formation of massive objects at high redshifts. Damped
Ly$\alpha$ systems are abundant in quasar absorption spectra and thus provide
possibly the most significant evidence for early structure formation, and thus
perhaps the most stringent constraint on CHDM. Using the numbers of halos in
N-body simulations to normalize Press-Schechter estimates of the number
densities of protogalaxies as a function of redshift, we find that CHDM with
$\Omega_c/\Omega_\nu/\Omega_b = 0.6/0.3/0.1$ is compatible with the damped
Ly$\alpha$ data at $\le 2.5$, but that it is probably incompatible with the
limited $z>3$ damped Ly$\alpha$ data. The situation is uncertain because there
is very little data for $z>3$, and also it is unclear whether all damped
Ly$\alpha$ systems are associated with collapsed protogalaxies. The predictions
of CHDM are quite sensitive to the hot (neutrino) fraction, and we find that
$\Omega_c/\Omega_\nu/\Omega_b = 0.675/0.25/0.075$ is compatible even with the
$z>3$ data. This corresponds to lowering the neutrino mass from 6.8 to 5.7 eV,
for $H_0=50\kmsMpc$. In CHDM, the higher redshift damped Ly$\alpha$ systems are
predicted to have lower masses, which can be checked by measuring the velocity
widths of the associated metal line systems.
