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Quasi-projective Moduli for Polarized Manifolds

  • Book
  • © 1995

Overview

Authors:
  1. Eckart Viehweg

    1. Fachbereich 6, Mathematik, Universität-Gesamthochschule Essen, Essen, Germany

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-VIII
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  2. Introduction

    • Eckart Viehweg
    Pages 1-14

  3. Moduli Problems and Hilbert Schemes

    • Eckart Viehweg
    Pages 15-52

  4. Weakly Positive Sheaves and Vanishing Theorems

    • Eckart Viehweg
    Pages 53-76

  5. D. Mumford’s Geometric Invariant Theory

    • Eckart Viehweg
    Pages 77-110

  6. Stability and Ampleness Criteria

    • Eckart Viehweg
    Pages 111-138

  7. Auxiliary Results on Locally Free Sheaves and Divisors

    • Eckart Viehweg
    Pages 139-166

  8. Weak Positivity of Direct Images of Sheaves

    • Eckart Viehweg
    Pages 167-196

  9. Geometric Invariant Theory on Hilbert Schemes

    • Eckart Viehweg
    Pages 197-238

  10. Allowing Certain Singularities

    • Eckart Viehweg
    Pages 239-276

  11. Moduli as Algebraic Spaces

    • Eckart Viehweg
    Pages 277-310

  12. Back Matter

    Pages 311-320
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Keywords

About this book

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.

Authors and Affiliations

  • Fachbereich 6, Mathematik, Universität-Gesamthochschule Essen, Essen, Germany

    Eckart Viehweg

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