Title Classifying spaces of degenerating polarized Hodge structures / Kazuya Kato and Sampei Usui

Published Princeton, N.J. : Princeton University Press, 2009

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Description 1 online resource (ix, 336 pages) : illustrations Series Annals of mathematics studies ; no. 169 Annals of mathematics studies ; no. 169. Contents 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 Summary In 1970, Philip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kato and Usui realize this dream by creating a logarithmic Hodge theory Bibliography Includes bibliographical references (pages 315-319) and index Notes Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002. http://purl.oclc.org/DLF/benchrepro0212 MiAaHDL English digitized 2011 HathiTrust Digital Library committed to preserve pda MiAaHDL Print version record Subject Hodge theory. Logarithms. logarithms. MATHEMATICS -- Topology. Hodge theory Logarithms Hodge-Struktur Hodge-Theorie Logarithmus Form Electronic book Author Usui, Sampei. LC no. 2008039091 ISBN 9780691138220 0691138222 9781400837113 1400837111

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