Table of Contents
Introduction 1
Chapter 0 Overview 7
0.1 Hodge Theory 7
0.2 Logarithmic Hodge Theory 11
0.3 Griffiths Domains and Moduli of PH 24
0.4 Toroidal Partial Compactifications of [Gamma]/D and Moduli of PLH 30
0.5 Fundamental Diagram and Other Enlargements of D 43
0.6 Plan of This Book 66
0.7 Notation and Convention 67
Chapter 1 Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits 70
1.1 Hodge Structures and Polarized Hodge Structures 70
1.2 Classifying Spaces of Hodge Structures 71
1.3 Extended Classifying Spaces 72
Chapter 2 Logarithmic Hodge Structures 75
2.1 Logarithmic Structures 75
2.2 Ringed Spaces (X[superscript log], [characters not reproducible]) 81
2.3 Local Systems on X[superscript log] 88
2.4 Polarized Logarithmic Hodge Structures 94
2.5 Nilpotent Orbits and Period Maps 97
2.6 Logarithmic Mixed Hodge Structures 105
Chapter 3 Strong Topology and Logarithmic Manifolds 107
3.1 Strong Topology 107
3.2 Generalizations of Analytic Spaces 115
3.3 Sets E[subscript sigma] and [characters not reproducible] 120
3.4 Spaces E[subscript sigma], [Gamma]/D[subscript Sigma], [characters not reproducible], and [characters not reproducible] 125
3.5 Infinitesimal Calculus and Logarithmic Manifolds 127
3.6 Logarithmic Modifications 133
Chapter 4 Main Results 146
4.1 Theorem A: The Spaces E[subscript sigma], [Gamma]/D[subscript Sigma], and [characters not reproducible] 146
4.2 Theorem B: The Functor [characters not reproducible] 147
4.3 Extensions of Period Maps 148
4.4 Infinitesimal Period Maps 153
Chapter 5 Fundamental Diagram 157
5.1 Borel-Serre Spaces (Review) 158
5.2 Spaces of SL(2)-Orbits (Review) 165
5.3 Spaces ofValuative Nilpotent Orbits 170
5.4 Valuative Nilpotent i-Orbits and SL(2)-Orbits 173
Chapter 6 The Map [psi]: [characters not reproducible] to D[subscript SL] (2) 175
6.1 Review of [CKS] and Some Related Results 175
6.2 Proof of Theorem 5.4.2 186
6.3 Proof of Theorem 5.4.3 (i) 190
6.4 Proofs of Theorem 5.4.3 (ii) and Theorem 5.4.4 195
Chapter 7 Proof of Theorem A 205
7.1 Proof of Theorem A (i) 205
7.2 Action of [sigma subscript C] on E[subscript sigma] 209
7.3 Proof of Theorem A for [Gamma]([sigma])[superscript gp]/D[subscript sigma] 215
7.4 Proof of Theorem A for [Gamma]/D[subscript Sigma] 220
Chapter 8 Proof of Theorem B 226
8.1 Logarithmic Local Systems 226
8.2 Proof of Theorem B 229
8.3 Relationship among Categories of Generalized Analytic Spaces 235
8.4 Proof of Theorem 0.5.29 241
Chapter 9 b-Spaces 244
9.1 Definitions and Main Properties 244
9.2 Proofs of Theorem 9.1.4 for [characters not reproducible], and [characters not reproducible] 246
9.3 Proof of Theorem 9.1.4 for [Gamma]/[characters not reproducible] 248
9.4 Extended Period Maps 249
Chapter 10 Local Structures of D[subscript SL(2)] and [Gamma]/[characters not reproducible] 251
10.1 Local Structures of D[subscript SL(2)] 251
10.2 A Special Open Neighborhood U(p) 255
10.3 Proof of Theorem 10.1.3 263
10.4 Local Structures of D[subscript SL(2). less than or equal 1] and [characters not reproducible] 269
Chapter 11 Moduli of PLH with Coefficients 271
11.1 Space [characters not reproducible] 271
11.2 PLH with Coefficients 274
11.3 Moduli 275
Chapter 12 Examples and Problems 277
12.1 Siegel Upper Half Spaces 277
12.2 Case G[subscript R] [characters not reproducible] O(1, n - 1, R) 281
12.3 Example of Weight 3 (A) 290
12.4 Example of Weight 3 (B) 295
12.5 Relationship with [U2] 299
12.6 Complete Fans 301
12.7 Problems 304
Appendix 307
A1 Positive Direction of Local Monodromy 307
A2 Proper Base Change Theorem for Topological Spaces 310
References 315
List of Symbols 321
Index 331
