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Quasi-projective Moduli for Polarized Manifolds


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Table of Contents

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.

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