  The role in string theory of manifolds of complex dimension $D_{crit} +
2(Q-1)$ and positive first Chern class is described. In order to be useful for
string theory, the first Chern class of these spaces has to satisfy a certain
relation. Because of this condition the cohomology groups of such manifolds
show a specific structure. A group that is particularly important is described
by $(D_{crit} + Q-1, Q-1)$--forms because it is this group which contains the
higher dimensional counterpart of the holomorphic $(D_{crit}, 0)$--form that
figures so prominently in Calabi--Yau manifolds. It is shown that the higher
dimensional manifolds do not, in general, have a unique counterpart of this
holomorphic form of rank $D_{crit}$. It is also shown that these manifolds
lead, in general, to a number of additional modes beyond the standard
Calabi--Yau spectrum. This suggests that not only the dilaton but also the
other massless string modes, such as the antisymmetric torsion field, might be
relevant for a possible stringy interpretation.
