  Heterotic compactifications on Calabi-Yau threefolds frequently exhibit
textures of vanishing Yukawa couplings in their low energy description. The
vanishing of these couplings is often not enforced by any obvious symmetry and
appears to be topological in nature. Recent results in the literature used
differential geometric methods to explain the origin of some of this structure.
A vanishing theorem was given which showed that the effect could be attributed,
in part, to the embedding of the Calabi-Yau manifolds of interest inside higher
dimensional ambient spaces, if the gauge bundles involved descended from vector
bundles on those larger manifolds. In this paper, we utilize an
algebro-geometric approach to provide an alternative derivation of some of
these results, and are thus able to generalize them to a much wider arena than
has been considered before. For example, we consider cases where the vector
bundles of interest do not descend from bundles on the ambient space. In such a
manner we are able to highlight the ubiquity with which textures of vanishing
Yukawa couplings can be expected to arise in heterotic compactifications, with
multiple different constraints arising from a plethora of different geometric
features associated to the gauge bundle.
