Quasi-Projective Moduli for Polarized Manifolds

This book discusses two subjects of quite different nature: Construction methods for quotients of quasi-projective schemes 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.

The concept of moduli goes back to B. Riemann, who shows in 68] that the isomorphism class of a Riemann surface of genus 9 2 depends on 3g - 3 parameters, which he proposes to name "moduli." A precise formulation of global moduli problems in algebraic geometry, the definition of moduli schemes or of algebraic moduli spaces for curves and for certain higher dimensional manifolds have only been given recently (A. Grothendieck, D. Mumford, see 59]), as well as solutions in some cases. It is the aim of this monograph to present methods which allow over a field of characteristic zero to construct certain moduli schemes together with an ample sheaf. Our main source of inspiration is D. Mumford's "Geometric In variant Theory." We will recall the necessary tools from his book 59] and prove the "Hilbert-Mumford Criterion" and some modified version for the stability of points under group actions. As in 78], a careful study of positivity proper ties of direct image sheaves allows to use this criterion to construct moduli as quasi-projective schemes for canonically polarized manifolds and for polarized manifolds with a semi-ample canonical sheaf."

The subject of the book are construction techniques for classifying spaces of projective manifolds and positivity results for direct image sheaves. Both together allow the proof of the central result, the existence of quasi-projective moduli schemes for a large class of manifolds. The book is of great interest for researchers and graduate students in algebraic geometry or number theory.