Locally mixed symmetric spaces / Bruce Hunt.

Format:

Book

Author/Creator:

Hunt, Bruce, 1958- author.

Series:

Springer monographs in mathematics.

Springer Monographs in Mathematics

Language:

English

Subjects (All):

Symmetric spaces.

Physical Description:

1 online resource (637 pages)

Edition:

1st ed.

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

What do the classification of algebraic surfaces, Weyl's dimension formula and maximal orders in central simple algebras have in common?All are related to a type of manifold called locally mixed symmetric spaces in this book.

Contents:

Intro

Introduction

Contents

List of Tables

1 Symmetric Spaces

1.1 Homogeneous Spaces

1.1.1 Invariant Connections

1.1.2 Compact Homogeneous Spaces

1.1.3 Complex Homogeneous Spaces

1.1.4 Projective Embeddings

1.1.5 Non-compact Homogeneous Spaces

1.2 Symmetric Spaces

1.2.1 Globally Symmetric Spaces

1.2.2 Isometries

1.2.3 Dualities

1.2.4 Locally Symmetric Spaces

1.2.5 Examples

1.2.6 Riemannian Symmetric Spaces

1.3 Classification of Symmetric Spaces

1.3.1 Symmetric Lie Algebras

1.3.2 Structure of Symmetric Spaces

1.4 Symmetric Subpairs and Totally Geodesic Subspaces

1.5 Hermitian Symmetric Spaces

1.5.1 Compact Hermitian Symmetric Spaces

1.5.2 Non-compact Hermitian Symmetric Spaces

1.5.3 The Exceptional Domains

1.5.4 Cayley Transforms

1.5.5 Boundary Components

1.5.6 Appendix: Siegel Domains

1.6 Examples

1.6.1 The Poincaré Plane

1.6.2 Hyperbolic Spaces

1.6.3 Some Symmetric Spaces Arising from Exceptional Groups

1.6.4 Symmetric Spaces Related to SU(4)

1.6.5 Hermitian Symmetric Spaces of Grassmann Type

1.6.6 Projective Planes

1.7 Satake Compactifications

1.7.1 Compactifications

1.7.2 Borel-Serre Compactification

1.7.3 The Compactification overlinecalP n of calP n=SLn(mathbbC)/SU(n)

1.7.4 Satake Compactifications

1.8 Morse Theory and Symmetric Spaces

1.8.1 Generalizations of Morse Theory

1.8.2 Applications of Morse Theory to Symmetric Spaces

1.8.3 The Space of Loops

2 Locally Symmetric Spaces

2.1 Arithmetic Quotients

2.1.1 Commensurability

2.1.2 Classification of Arithmetic Groups (Examples)

2.2 Rational Boundary Components

2.2.1 The theorem of Gauß-Bonnet for Arithmetic Quotients

2.3 Compactifications of Arithmetic Quotients

2.3.1 Borel-Serre Compactification

2.3.2 Satake Compactifications.

2.4 Locally Hermitian Symmetric Spaces

2.4.1 Rational Boundary Components

2.4.2 Baily-Borel Embedding

2.4.3 Toroidal Compactifications of Locally Hermitian Symmetric Varieties

2.5 The Proportionality Principle

2.5.1 Hirzebruch Proportionality in the Non-compact Case

2.6 Locally Symmetric Subspaces

Totally Geodesic Subspaces

2.6.1 Geodesic Cycles

2.6.2 Non-vanishing (Co-)Homology

2.6.3 Relative Proportionality

2.7 Examples

2.7.1 Spaces Deriving from Geometric Forms

2.7.2 The Poincaré Plane

2.7.3 Hyperbolic 3-Folds

2.7.4 Picard Modular Varieties (Arithmetic Quotients of Complex Hyperbolic Manifolds)

2.7.5 Hyperbolic D-Planes

2.7.6 Arithmetic Quotients of Hermitian Symmetric Spaces of Grassmann Type

2.7.7 Janus-Like Algebraic Varieties

2.8 Locally Semisimple Symmetric Spaces

3 Locally Mixed Symmetric Spaces

3.1 Mixed Symmetric Spaces

3.1.1 Mixed Symmetric Pairs

3.1.2 Morphisms of Mixed Symmetric Pairs

3.1.3 Extensions of Mixed Symmetric Spaces to Compactifications

3.2 Locally Mixed Symmetric Spaces

3.2.1 Structure of the Fiber

3.3 Examples

3.3.1 Examples Deriving from Geometric Forms

3.3.2 Examples Arising from Exceptional Groups

3.4 Locally Mixed Symmetric Spaces and Compactifications

3.4.1 LMSS and the Borel-Serre Compactification

3.4.2 Embedding Locally Symmetric Spaces in Larger Ones

3.5 Global Sections

4 Kuga Fiber Spaces

4.1 Period Domains

4.1.1 Hodge Structures

4.1.2 Variation of Hodge Structures

4.1.3 Monodromy

4.1.4 Hodge Structures of Weight 2

4.2 Hodge Structures of Weight 1

4.2.1 Complex Tori

4.2.2 Siegel Spaces

4.2.3 Families of Abelian Varieties

4.3 Kuga Fiber Spaces

4.3.1 LMSS of Hermitian Type

4.3.2 Kuga Fiber Spaces

4.3.3 Polarized Hodge Structures of Weight 1.

4.3.4 Characterization of Kuga Fiber Spaces

4.4 Symplectic Representations of mathbbQ-Groups

4.4.1 Hermitian Forms, Symplectic Forms and Involutions

4.4.2 Holomorphic Embeddings of Symmetric Domains into a Siegel Space

4.4.3 Classification of Kuga Fiber Spaces

4.5 Pel Structures and Equivariant Embeddings

4.6 Modular Subvarieties, Boundary Components and Degenerations

4.6.1 Decompositions

4.6.2 Degenerations

4.6.3 Namikawa's Compactification

4.7 Examples

4.7.1 Hodge Structures of Weight 2

4.7.2 Families of Abelian Varieties with Real Multiplication

4.7.3 Families of Abelian Varieties with Complex Multiplication

4.7.4 Families of Abelian Varieties with Quaternion Multiplication

4.7.5 Hyperbolic D-Planes

4.7.6 A Ball Quotient Related to a Division Algebra

4.8 Group of Sections

5 Elliptic Surfaces

5.1 Elliptic Curves

5.2 Elliptic Surfaces

5.3 Singular Fibers

5.4 Homological and Functional Invariants

5.5 The Family of Elliptic Surfaces with Given Invariants

5.6 Numerical Invariants of Elliptic Surfaces

5.7 The Exponential Sequence

5.8 Elliptic Modular Surfaces

5.9 The Classifying Map of an Elliptic Surface

5.10 Weierstraß Models

5.11 Deformations and Moduli

5.12 Appendix: Curves on a Compact Complex Surface

6 Appendices

6.1 Algebra

6.1.1 Geometric Forms

6.1.2 K-Algebras

6.1.3 Division Algebras

6.2 Topology and Differential Geometry

6.2.1 Homotopy, Classifying Spaces and Fiber Bundles

6.2.2 Leray-Hirsch Theorem

6.2.3 Characteristic Classes

6.2.4 Differential Geometry

6.2.5 Lie Groups and Lie Algebras

6.3 Complex Geometry and Algebraic Groups

6.3.1 Complex Manifolds and Algebraic Varieties

6.3.2 Hodge Structures

6.3.3 Abelian Varieties

6.3.4 Algebraic Groups

6.3.5 Arithmetic Groups.

6.4 Exceptional Algebraic and Lie Groups

6.4.1 Real Lie Groups

6.4.2 Octonions

6.4.3 Jordan Algebras

6.4.4 Exceptional Lie Algebras

6.5 Some Finite Geometry

6.5.1 Isotropic Subspaces

6.5.2 Non-degenerate Subspaces

6.5.3 The Index of P Γg(N) in PSp2g(mathbbZ)

Appendix References

Index.

Notes:

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

3-030-69804-1

OCLC:

1267456599