Quasi-projective Moduli for Polarized Manifolds by Eckart Viehweg
Quasi-projective Moduli for Polarized Manifolds - Table of Contents
1. Moduli Problems and Hilbert Schemes
2. Weakly Positive Sheaves and Vanishing Theorems
3. D. Mumford’s Geometric Invariant Theory
4. Stability and Ampleness Criteria
5. Auxiliary Results on Locally Free Sheaves and Divisors
6. Weak Positivity of Direct Images of Sheaves
7. Geometric Invariant Theory on Hilbert Schemes
8. Allowing Certain Singularities
9. Moduli as Algebraic Spaces
What You Will Learn in Quasi-projective Moduli for Polarized Manifolds
"Quasi-projective Moduli for Polarized Manifolds" by "Eckart Viehweg" is a classic research monograph in modern algebraic geometry that studies how to construct well-behaved moduli spaces for higher-dimensional algebraic varieties. The book focuses on "polarized manifolds", meaning smooth projective varieties equipped with an ample line bundle, and addresses the central question of when their moduli spaces can be realized as "quasi-projective varieties".
The core contribution of the book is a systematic method for proving quasi-projectivity using tools such as "Geometric Invariant Theory (GIT)", Hilbert schemes, and positivity properties of direct image sheaves. Viehweg develops the theory of "weak positivity", which plays a crucial role in constructing ample line bundles on moduli spaces. These techniques allow one to move from abstract moduli functors to concrete geometric spaces that parametrize families of polarized manifolds.
A major result is the existence of quasi-projective coarse moduli spaces for classes of manifolds with semi-ample or ample canonical bundles, including canonically polarized varieties. Written for advanced graduate students and researchers, the book has had lasting influence on "moduli theory", higher-dimensional algebraic geometry, and later work connecting algebraic geometry with complex geometry.
Book Details & Specifications
Title:
Quasi-projective Moduli for Polarized Manifolds by Eckart Viehweg
Publisher:
Springer
Year:
1995
Pages:
326
Type:
PDF
Language:
English
ISBN-10 #:
3642797474
ISBN-13 #:
978-3642797477
License:
Made available online with permission from Springer-Verlag
Amazon:
Amazon
About the Author: Eckart Viehweg
The author Eckart Viehweg
was a leading mathematician in "algebraic geometry", widely recognized for his foundational work in "moduli theory" and complex geometry. His research focused on the structure of "quasi-projective varieties", positivity techniques, and higher-dimensional geometry, shaping modern approaches to moduli spaces. In "Quasi-projective Moduli for Polarized Manifolds", Viehweg applies deep theoretical insight to the study of "polarized manifolds" and their moduli. His writing reflects mathematical rigor and long-lasting influence, making the work a cornerstone for researchers exploring "complex algebraic varieties" and geometric classification theory.
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