Coherent Sheaves on Calabi-Yau manifolds Picard-Fuchs equations and potential functions

We investigate the deformation theory of Calabi-Yau threefolds -- simply connected complex projective manifolds with trivial canonical bundle -- together with geometric objects on the Calabi-Yau threefold. The geometric objects in question are curves, surfaces, special coherent sheaves and rank-two vector bundles with a section vanishing in codimension two. These deformation problems are related to the study of Picard-Fuchs equations. Physicists suggest that a holomorphic potential function the critical locus of which is the space of unobstructed deformations appears as a solution of the Picard-Fuchs equation. In this thesis, Picard-Fuchs equations for Calabi-Yau threefolds appearing as complete intersections of codimension two in projective spaces are studied. In addition, Picard-Fuchs equations for pairs of a Calabi-Yau threefold and either a divisor or a curve on the threefold are examined. We construct Picard-Fuchs equations using Griffiths-Dwork reduction. Based on the work of Jockers and Soroush, we give rigorous mathematical foundations for deriving Picard-Fuchs operators in various cases, in particular for the quintic threefold together with a special divisor. Furthermore, we initiate a theory of triples consisting of a threefold with two divisors meeting transversally along a curve and set up Picard-Fuchs operators for this situation. The thesis furthermore contains some SINGULAR programmes for explicit calculations.