3 Review(s)

This text discusses two subjects of quite different natures: construction methods for quotients of quasi-projective schemes either by group actions or by equivalence relations; and properties of direct images of certain sheaves under smooth morphisms. Both methods together allow to prove the central result of the text, the existence of quasi-projective moduli schemes, whose points parametrize the set of manifolds with ample canonical divisors or the set of polarized manifolds with a semi-ample canonical divisor. Starting with A. Grothendieck's construction of Hibert schemes, including the basics of D. Mumford's geometric invariant theory and an introduction to M. Artin's theory of algebraic spaces, the reader finds the tools for the construction of moduli, usually not contained in textbooks on algebraic geometry.