  We consider superstring compactifications where both the classical
description, in terms of a Calabi-Yau manifold, and also the quantum theory is
known in terms of a Landau-Ginzburg orbifold model. In particular, we study
(smooth) Calabi-Yau examples in which there are obstructions to parametrizing
all of the complex structure cohomology by polynomial deformations thus
requiring the analysis based on exact and spectral sequences. General arguments
ensure that the Landau-Ginzburg chiral ring copes with such a situation by
having a nontrivial contribution from twisted sectors. Beyond the expected
final agreement between the mathematical and physical approaches, we find a
direct correspondence between the analysis of each, thus giving a more complete
mathematical understanding of twisted sectors. Furthermore, this approach shows
that physical reasoning based upon spectral flow arguments for determining the
spectrum of Landau-Ginzburg orbifold models finds direct mathematical
justification in Koszul complex calculations and also that careful point- field
analysis continues to recover suprisingly much of the stringy features.
