Moduli Theory and Classification Theory of Algebraic Varieties

Moduli theory of algebraic varieties and classification theory of compact complex spaces.- Moduli spaces for polarized algebraic varieties.- Group quotients in the category of analytic spaces and the category of algebraic spaces.- Applications of the quotient theorems to moduli of algebraic varieties.- Quotients for affine schemes by reductive algebraic groups.- Quotients in the category of schemes.- Mumford¿s construction of the moduli variety for curves and polarized abelian varieties. Other applications of Mumford¿s quotient theory.- Other methods of treating moduli problems. Artin¿s method of algebraic stacks. Griffiths¿s method of period maps.- Compactification of moduli spaces.- Fine moduli spaces. The universal families for stable curves with level n-structure.- Applications of moduli theory to fibre spaces and the additivity formula for the Kodaira dimension of fibre spaces. Open problems.