We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semi jcal approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some “extra structure” described by means of es in H$^2$. We formulate a simple and compelling conjecture about the action of the automorphism group on the Kähler cone, which would enable the construction of a partial compactification of the moduli space using Looijenga's “semi-toric” method. We then 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.