Classifying Spaces of Degenerating Polarized Hodge Structures
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A01=Kazuya Kato
A01=Sampei Usui
Algebraic group
Algebraic variety
Analytic manifold
Analytic space
Annulus (mathematics)
Arithmetic group
Atlas (topology)
Author_Kazuya Kato
Author_Sampei Usui
Canonical map
Category=PBKF
Classifying space
Coefficient
Cohomology
Compactification (mathematics)
Complex manifold
Complex number
Congruence subgroup
Continuous function
Convex cone
Degeneracy (mathematics)
Diagram (category theory)
Differential form
Direct image functor
Divisor
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equivalence class
Existential quantification
Geometry
Hodge structure
Hodge theory
Homeomorphism
Homomorphism
Inverse function
Iwasawa decomposition
Local homeomorphism
Local ring
Local system
Maximal compact subgroup
Modular curve
Modular form
Moduli space
Monodromy
Monoid
Morphism
Nilpotent
Nilpotent orbit
Open set
P-adic Hodge theory
P-adic number
Point at infinity
Proper morphism
Pullback (category theory)
Quotient space (topology)
Relative interior
Ring (mathematics)
Ring homomorphism
Set (mathematics)
Sheaf (mathematics)
Smooth morphism
Special case
Strong topology
Subgroup
Subobject
Subset
Surjective function
Tangent bundle
Taylor series
Theorem
Topology
Transversality (mathematics)
Two-dimensional space
Vector bundle
Vector space
Product details
- ISBN 9780691138220
- Weight: 595g
- Dimensions: 152 x 235mm
- Publication Date: 07 Dec 2008
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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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.
Kazuya Kato is professor of mathematics at Kyoto University. Sampei Usui is professor of mathematics at Osaka University.
