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 Γ\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 (χlog,Qlogx) ......................... 81
2.3. Local Systems on χlog ............................. 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 Eσ and Eσ# .................................. 120
3.4. Spaces Eσ, Γ\D∑ Eσ# and D∑# ....................... 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 Eσ, Γ\D∑ and Γ\D∑# ......... 146
4.2. Theorem B: The Functor PLHΦ ...................... 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 of Valuative Nilpotent Orbits ............. 170
5.4. Valuative Nilpotent i-Orbits and SL(2)-Orbits .... 173
Chapter 6. The Map ψ: D#val → DSl(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 σC on Eσ ............................... 209
7.3. Proof of Theorem A for Γ(σ)gp\Dσ ................. 215
7.4. ProofofTheorem Afor Γ\D∑ ......................... 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 Γ\χbBS, Γ\DbBS and
Γ\DbBS.val .............................. 246
9.3. Proof of Theorem 9.1.4 for Γ\DbSL(2). ≤ 1 ......... 248
9.4. Extended Period Maps ............................. 249
Chapter 10.Local Structures of DSL(2) and Γ\DbSL(2) ≤ 1 ......... 251
10.1.Local Structures of DSL(2) ........................ 251
10.2.A Special Open Neighborhood U{p) ................. 255
10.3.Proof of Theorem 10.I.3. ......................... 263
10.4.Local Structures of DSL(2). ≤ 1 and Γ/DbSL(2)≤1 ..... 269
Chapter 11.Moduli of PLH with Coefficients .................... 271
11.1.Space Γ\DA∑ ...................................... 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 CR ≈ 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
Al Positive Direction of Local Monodromy ...................... 307
A2 Proper Base Change Theorem for Topological Spaces .......... 310
References .................................................... 315
List of Symbols ............................................... 321
Index ......................................................... 331
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