  We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburg
correspondence to a more general class of manifolds. Specifically we consider
the Fermat type hypersurfaces $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bf
CP}^{N-1}$ for various values of k and N. When k<N, the 1-loop beta function of
the sigma model on $M_N^k$ is negative and we expect the theory to have a mass
gap. However, the quantum cohomology relation
$\sigma^{N-1}={const.}\sigma^{k-1}$ suggests that in addition to the massive
vacua there exists a remaining massless sector in the theory if k>2. We assume
that this massless sector is described by a Landau-Ginzburg (LG) theory of
central charge $c=3N(1-2/k)$ with N chiral fields with U(1) charge $1/k$. We
compute the topological invariants (elliptic genera) using LG theory and
massive vacua and compare them with the geometrical data. We find that the
results agree if and only if k=even and N=even.
  These are the cases when the hypersurfaces have a spin structure. Thus we
find an evidence for the geometry/LG correspondence in the case of spin
manifolds.
