Considerations of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps

We consider Calabi-Yau compactifications with one Kähler modulus. Following the method of Candelas et al. we use the mirror hypotheses to solve the quantum theory exactly in dependence of this modulus by performing the calculation for the corresponding complex structure deformation on the mirror manifold. Here the information is accessible by techniques of classical geometry. It is encoded in the Picard-Fuchs differential equation which has to be supplemented by requirements on the global properties of its solutions