  We investigate a Gepner-like superstring model described by a combination of
multiple minimal models and an N=2 Liouville theory. This model is thought to
be equivalent to the superstring theory on a singular noncompact Calabi-Yau
manifold. We construct the modular invariant partition function of this model,
and confirm the validity of an appropriate GSO projection. We also calculate
the elliptic genus and Witten index of the model. We find that the elliptic
genus factorises into a rather trivial factor and a non-trivial one, and the
non-trivial one has the information on the positively curved base manifold of
the cone.
