Advances in moduli theory

This book aims to study several aspects of moduli theory from a complex analytic point of view. Chapter 1 gives a brief introduction to the Kodaira-Spencer deformation theory of compact complex manifolds. In Chapter 2 we discuss the analytic theory of abelian varieties, and show how to construct the moduli theory of polarized abelian varieties. Also the moduli of closed Riemann surfaces and Torelli's theorem will be discussed. Chapter 3 gives the basics of Hodge theory. In the final chapter, as an application of the moduli theory of curves we shall discuss non-abelian conformal field theory as formulated by Tsuchiya, Ueno, and Yamada. Its relation to the moduli of vector bundles on a closed Riemann surface will also be discussed. The book is aimed at graduate and upper-level undergraduate students who want to learn modern moduli theory.