  We consider generalized complete intersection manifolds in the product space
of projective spaces, and work out useful aspects pertaining to the cohomology
of sheaves over them. First, we present and prove a vanishing theorem on the
cohomology groups of sheaves for subvarieties of the ambient product space of
projective spaces. We then prove an equivalence between configuration matrices
of complete intersection Calabi-Yau manifolds. We also present a formula of the
genus of curves in generalized complete intersection manifolds. Some of these
curves arise as the fixed point locus of certain symmetry group action on the
generalized complete intersection Calabi-Yau manifolds. We also make a
blowing-up along the curves, by which one can generate new Calabi-Yau
manifolds. Moreover, an approach on spectral sequences is used to compute Hodge
numbers of generalized complete intersection Calabi-Yau manifolds and the genus
of curves therein.
