Classifying Spaces of Degenerating Polarized Hodge Structures

Description

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 revive nilpotent orbits as a logarithmic Hodge structure. The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinement of the toroidal compactifications by Mumford et al. For general D, fine moduli spaces may have slits caused by Griffiths transversality at the boundary and be no longer locally compact. Second, Kato and Usui construct eight enlargements of D and describe their relations by a fundamental diagram, where four of these enlargements live in the Hodge theoretic area and the other four live in the algebra-group theoretic area. These two areas are connected by a continuous map given by the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the construction in the first topic. Contents......Page 7 Introduction......Page 13 0. Overview......Page 19 1. Spaces of Nilpotent Orbits and Spaces of Nilpotent i-Orbits......Page 82 2. Logarithmic Hodge Structures......Page 87 3. Strong Topology and Logarithmic Manifolds......Page 119 4. Main Results......Page 158 5. Fundamental Diagram......Page 169 6. The Map ψ......Page 187 7. Proof of Theorem A......Page 217 8. Proof of Theorem B......Page 238 9. b-Spaces......Page 256 10. Local Structures of D_SL(2) and......Page 263 11. Moduli of PLH with Coefficients......Page 283 12. Examples and Problems......Page 289 Appendix......Page 319 References......Page 327 List of Symbols......Page 333 Index......Page 343