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

TL;DR

This paper explores the cohomology of sheaves on generalized complete intersection Calabi-Yau manifolds, providing vanishing theorems, configuration equivalences, genus formulas, and methods to compute Hodge numbers and generate new manifolds.

Contribution

It introduces new vanishing theorems, configuration matrix equivalences, and a spectral sequence approach for computing Hodge numbers in generalized Calabi-Yau manifolds.

Findings

Vanishing theorem for sheaf cohomology groups on subvarieties.

Equivalence between different configuration matrices of Calabi-Yau manifolds.

Formula for the genus of curves in these manifolds.

Abstract

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.