| Superconformal symmetry play a fundamental role in string theory. It is also deeply related to Kähler structures on complex manifolds and the moonshine module. We shall begin with some basic material in complex geometry, including complex manifolds, sheaf cohomology, Hermitian metrics, holomorphic vector bundles, connections, Hodge theory for complex manifolds, Kähler manifolds, Chern classes and Calabi-Yau manifolds. We shall study N=1 and N=2 superconformal algebras and their representations. Vertex operator superalgebras and modules associated to these representations will be constructed and studied and their "world-sheet" geometry will be discussed. We shall also discuss Gepner models and the conjectured N=2 superconformal field theories associated to Calabi-Yau manifolds. |