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.

Inhalt

Leitfaden.- Classification Theory and Moduli Problems.- Notations and Conventions.- 1 Moduli Problems and Hilbert Schemes.- 1.1 Moduli Functors and Moduli Schemes.- 1.2 Moduli of Manifolds: The Main Results.- 1.3 Properties of Moduli Functors.- 1.4 Moduli Functors for ?-Gorenstein Schemes.- 1.5 A. Grothendieck's Construction of Hilbert Schemes.- 1.6 Hilbert Schemes of Canonically Polarized Schemes.- 1.7 Hilbert Schemes of Polarized Schemes.- 2 Weakly Positive Sheaves and Vanishing Theorems.- 2.1 Coverings.- 2.2 Numerically Effective Sheaves.- 2.3 Weakly Positive Sheaves.- 2.4 Vanishing Theorems and Base Change.- 2.5 Examples of Weakly Positive Sheaves.- 3 D. Mumford's Geometric Invariant Theory.- 3.1 Group Actions and Quotients.- 3.2 Linearizations.- 3.3 Stable Points.- 3.4 Properties of Stable Points.- 3.5 Quotients, without Stability Criteria.- 4 Stability and Ampleness Criteria.- 4.1 Compactifications and the Hilbert-Mumford Criterion.- 4.2 Weak Positivity of Line Bundles and Stability.- 4.3 Weak Positivity of Vector Bundles and Stability.- 4.4 Ampleness Criteria.- 5 Auxiliary Results on Locally Free Sheaves and Divisors.- 5.1 O. Gabber's Extension Theorem.- 5.2 The Construction of Coverings.- 5.3 Singularities of Divisors.- 5.4 Singularities of Divisors in Flat Families.- 5.5 Vanishing Theorems and Base Change, Revisited.- 6 Weak Positivity of Direct Images of Sheaves.- 6.1 Variation of Hodge Structures.- 6.2 Weakly Semistable Reduction.- 6.3 Applications of the Extension Theorem.- 6.4 Powers of Dualizing Sheaves.- 6.5 Polarizations, Twisted by Powers of Dualizing Sheaves.- 7 Geometric Invariant Theory on Hilbert Schemes.- 7.1 Group Actions on Hilbert Schemes.- 7.2 Geometric Quotients and Moduli Schemes.- 7.3 Methods to Construct Quasi-Projective Moduli Schemes.- 7.4 Conditions for the Existence of Moduli Schemes: Case (CP).- 7.5 Conditions for the Existence of Moduli Schemes: Case (DP).- 7.6 Numerical Equivalence.- 8 Allowing Certain Singularities.- 8.1 Canonical and Log-Terminal Singularities.- 8.2 Singularities of Divisors.- 8.3 Deformations of Canonical and Log-Terminal Singularities.- 8.4 Base Change and Positivity.- 8.5 Moduli of Canonically Polarized Varieties.- 8.6 Moduli of Polarized Varieties.- 8.7 Towards Moduli of Canonically Polarized Schemes.- 9 Moduli as Algebraic Spaces.- 9.1 Algebraic Spaces.- 9.2 Quotients by Equivalence Relations.- 9.3 Quotients in the Category of Algebraic Spaces.- 9.4 Construction of Algebraic Moduli Spaces.- 9.5 Ample Line Bundles on Algebraic Moduli Spaces.- 9.6 Proper Algebraic Moduli Spaces for Curves and Surfaces.- References.- Glossary of Notations.