Classifying Spaces of Degenerating Polarized Hodge Structures. (AM-169)
In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to...
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| Format: | eBook |
| Language: | English |
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Princeton :
Princeton University Press,
2008.
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| Series: | Annals of mathematics studies ;
no. 169. |
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Table of Contents:
- Cover; Title; Copyright; Contents; Introduction; Chapter 0. Overview; 0.1 Hodge Theory; 0.2 Logarithmic Hodge Theory; 0.3 Griffiths Domains and Moduli of PH; 0.4 Toroidal Partial Compactifications of Г\D and Moduli of PLH; 0.5 Fundamental Diagram and Other Enlargements of D; 0.6 Plan of This Book; 0.7 Notation and Convention; Chapter 1. Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits; 1.1 Hodge Structures and Polarized Hodge Structures; 1.2 Classifying Spaces of Hodge Structures; 1.3 Extended Classifying Spaces; Chapter 2. Logarithmic Hodge Structures; 2.1 Logarithmic Structures.
- 2.2 Ringed Spaces (X̂log, Ôlog X)2.3 Local Systems on X̂log; 2.4 Polarized Logarithmic Hodge Structures; 2.5 Nilpotent Orbits and Period Maps; 2.6 Logarithmic Mixed Hodge Structures; Chapter 3. Strong Topology and Logarithmic Manifolds; 3.1 Strong Topology; 3.2 Generalizations of Analytic Spaces; 3.3 Sets Eσ and Ê♯σ; 3.4 Spaces Eσ, Г\DΣ, Ê♯σ, and D̂♯Σ; 3.5 Infinitesimal Calculus and Logarithmic Manifolds; 3.6 Logarithmic Modifications; Chapter 4. Main Results; 4.1 Theorem A: The Spaces Eσ, Г\DΣ and Г\DΣ♯; 4.2 Theorem B: The Functor PLНФ; 4.3 Extensions of Period Maps.
- 4.4 Infinitesimal Period MapsChapter 5. Fundamental Diagram; 5.1 Borel-Serre Spaces (Review); 5.2 Spaces of SL(2)-Orbits (Review); 5.3 Spaces of Valuative Nilpotent Orbits; 5.4 Valuative Nilpotent i-Orbits and SL(2)-Orbits; Chapter 6. The Map ψ : D̂♯val → DSL(2); 6.1 Review of [CKS] and Some Related Results; 6.2 Proof of Theorem 5.4.2; 6.3 Proof of Theorem 5.4.3 (i); 6.4 Proofs of Theorem 5.4.3 (ii) and Theorem 5.4.4; Chapter 7. Proof of Theorem A; 7.1 Proof of Theorem A (i); 7.2 Action of σC on Eσ; 7.3 Proof of Theorem A for Г(σ)̂gp\Dσ; 7.4 Proof of Theorem A for Г\DΣ
- Chapter 8. Proof of Theorem B8.1 Logarithmic Local Systems; 8.2 Proof of Theorem B; 8.3 Relationship among Categories of Generalized Analytic Spaces; 8.4 Proof of Theorem 0.5.29; Chapter 9. ♭-Spaces; 9.1 Definitions and Main Properties; 9.2 Proofs of Theorem 9.1.4 for Г\X̂♭BS, Г\D̂♭BS, and Г\D̂♭BS, val; 9.3 Proof of Theorem 9.1.4 for Г\D̂♭SL(2), ≤1; 9.4 Extended Period Maps; Chapter 10. Local Structures of DSL(2) and Г\D̂♭SL(2), ≤1; 10.1 Local Structures of DSL(2); 10.2 A Special Open Neighborhood U(p); 10.3 Proof of Theorem 10.1.3; 10.4 Local Structures of DSL(2), ≤1 and Г\D̂♭SL(2), ≤1.
- Chapter 11. Moduli of PLH with Coefficients11.1 Space Г\ D̂AΣ; 11.2 PLH with Coefficients; 11.3 Moduli; Chapter 12. Examples and Problems; 12.1 Siegel Upper Half Spaces; 12.2 Case GR ≃ O(1, n − 1, R); 12.3 Example of Weight 3 (A); 12.4 Example of Weight 3 (B); 12.5 Relationship with [U2]; 12.6 Complete Fans; 12.7 Problems; Appendix; A1 Positive Direction of Local Monodromy; A2 Proper Base Change Theorem for Topological Spaces; References; List of Symbols; Index.