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We consider generalized complete intersection manifolds in the product spaceof projective spaces, and work out useful aspects pertaining to the cohomologyof sheaves over them. First, we present and prove a vanishing theorem on thecohomology groups of sheaves for subvarieties of the ambient product space ofprojective spaces. We then prove an equivalence between configuration matricesof complete intersection Calabi-Yau manifolds. We also present a formula of thegenus of curves in generalized complete intersection manifolds. Some of thesecurves arise as the fixed point locus of certain symmetry group action on thegeneralized complete intersection Calabi-Yau manifolds. We also make ablowing-up along the curves, by which one can generate new Calabi-Yaumanifolds. Moreover, an approach on spectral sequences is used to compute Hodgenumbers of generalized complete intersection Calabi-Yau manifolds and the genusof curves therein.

Calabi-Yau generalized complete intersections and aspects of cohomology of sheaves