  We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds,
using the one-loop semiclassical approximation. The data being parameterized
includes a choice of complex structure on the manifold, as well as some ``extra
structure'' described by means of classes in H^2. The expectation that this
moduli space is well-behaved in these ``extra structure'' directions leads us
to formulate a simple and compelling conjecture about the action of the
automorphism group on the K\"ahler cone. If true, it allows one to apply
Looijenga's ``semi-toric'' technique to construct a partial compactification of
the moduli space. We explore the implications which this construction has
concerning the properties of the moduli space of complex structures on a
``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a
similarity which might have been noticed between certain work of Mumford and of
Mori from the 1970's produces (with hindsight) evidence for mirror symmetry
which was available in 1979. [The author is willing to mail hardcopy preprints
upon request.]
