Classifying spaces of degenerating polarized Hodge structures

 Content: 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 --
 0.7 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], O[subscript X superscript log]) 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 E[subscript sigma superscript sharp] 120 --
 3.4 Spaces E[subscript sigma], [Gamma]/D[subscript Sigma], E[subscript sigma superscript sharp], and D[subscript Sigma superscript sharp] 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 [Gamma]/D[subscript Sigma sharp] 146 --
 4.2 Theorem B: The Functor PLH[subscript phi] 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 [psi] : D[subscript val superscript sharp] 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 [flat]-Spaces 244 --
 9.1 Definitions and Main Properties 244 --
 9.2 Proofs of Theorem 9.1.4 for [Gamma]/X[subscript BS superscript flat], [Gamma]/D[superscript flat subscript BS], and [Gamma]/D[subscript BS, val superscript flat] 246 --
 9.3 Proof of Theorem 9.1.4 for [Gamma]/D[subscript SL(2), less than or equal 1 superscript flat] 248 --
 9.4 Extended Period Maps 249 --
 Chapter 10 Local Structures of D[subscript SL(2)] and [Gamma]/D[subscript SL(2), less than or equal 1 superscript flat] 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 [Gamma]/D[subscript SL(2), less than or equal 1 superscript flat] 269 --
 Chapter 11 Moduli of PLH with Coefficients 271 --
 11.1 Space [Gamma]/D[subscript Sigma superscript A] 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] [bsime] 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 --
 A1 Positive Direction of Local Monodromy 307 --
 A2 Proper Base Change Theorem for Topological Spaces 310.
 Abstract:             
              In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. This book realizes this by creating a logarithmic Hodge theory. It uses the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure.                 Read more...