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Tables of Contents for Period Mappings and Period Domains
Chapter/Section Title
Page #
Page Count
Preface
xiii
Part I: Basic Theory of the Period Map
1 Introductory Examples
3
68
1.1 Elliptic Curves
3
22
1.2 Riemann Surfaces of Higher Genus
25
22
1.3 Double Planes
47
15
1.4 Mixed Hodge Theory Revisited
62
9
2 Cohomology of Compact Kähler Manifolds
71
23
2.1 Cohomology of Compact Differentiable Manifolds
71
6
2.2 What Happens on Kähler Manifolds
77
11
2.3 How Lefschetz Further Decomposes Cohomology
88
6
3 Holomorphic Invariants and Cohomology
94
38
3.1 Is the Hodge Decomposition Holomorphic?
94
9
3.2 A Case Study: Hypersurfaces
103
10
3.3 How Log-Poles Lead to Mixed Hodge Structures
113
5
3.4 Algebraic Cycles and Their Cohomology Classes
118
6
3.5 Tori Associated to Cohomology
124
3
3.6 Abel-Jacobi Maps
127
5
4 Cohomology of Manifolds Varying in a Family
132
27
4.1 Smooth Families and Monodromy
132
3
4.2 An Example: Lefschetz Theory
135
4
4.3 Variations of Hodge Structures Make Their First Appearance
139
8
4.4 Period Maps
147
7
4.5 Abstract Variations of Hodge Structure
154
2
4.6 The Abel-Jacobi Map Revisited
156
3
5 Period Maps Looked at Infinitesimally
159
22
5.1 Deformations of Compact Complex Manifolds Over a Smooth Base
159
4
5.2 Enter: The Thick Point
163
3
5.3 The Derivative of the Period Map
166
3
5.4 An Example: Deformations of Hypersurfaces
169
3
5.5 Infinitesimal Variations of Hodge Structure
172
4
5.6 Application: A Criterion for the Period Map to Be an Immersion
176
5
Part II: The Period Map: Algebraic Methods
6 Spectral Sequences
181
12
6.1 Fundamental Notions
181
3
6.2 Hypercohomology Revisited
184
2
6.3 de Rham Theorems
186
2
6.4 The Hodge Filtration Revisited
188
5
7 Koszul Complexes and Some Applications
193
25
7.1 The Basic Koszul Complexes
193
4
7.2 Koszul Complexes of Sheaves on Projective Space
197
3
7.3 Castelnuovo's Regularity Theorem
200
6
7.4 Macaulay's Theorem and Donagi's Symmetrizer Lemma
206
5
7.5 Applications: The Noether-Lefschetz Theorems
211
7
8 Further Applications: Torelli Theorems for Hypersurfaces
218
24
8.1 Infinitesimal Torelli Theorems
218
5
8.2 Variational and Generic Torelli
223
2
8.3 Global Torelli for Hypersurfaces
225
6
8.4 Moduli
231
11
9 Normal Functions and Their Applications
242
23
9.1 Normal Functions and Infinitesimal Invariants
242
7
9.2 The Infinitesimal Invariant as a Relative Cycle Class
249
6
9.3 Primitive (p, p)-Classes and the Griffiths Group of Hypersurface Sections
255
5
9.4 The Theorem of Green and Voisin
260
5
10 Applications to Algebraic Cycles: Non's Theorem
265
38
10.1 A Detour into Deligne Cohomology with Applications
265
4
10.2 The Statement of Non's Theorem
269
5
10.3 A Local-to-Global Principle
274
3
10.4 Jacobian Representations Revisited
277
7
10.5 A Proof of Non's Theorem
284
5
10.6 Applications of Non's Theorem and Filtrations on Chow Groups
289
14
Part III: Differential Geometric Methods
11 Further Differential Geometric Tools
303
16
11.1 Chern Connections and Applications
303
4
11.2 Subbundles and Quotient Bundles
307
4
11.3 Principal Bundles and Connections
311
8
12 Structure of Period Domains
319
16
12.1 Homogeneous Bundles on Homogeneous Spaces
319
6
12.2 The Lie Algebra Structure of Groups Defining Period Domains
325
4
12.3 Canonical Connections on Reductive Spaces
329
6
13 Curvature Estimates and Applications
335
24
13.1 Curvature of Hodge Bundles
336
10
13.2 Curvature Bounds over Compact Curves
346
3
13.3 Curvature of Period Domains
349
3
13.4 Applications
352
7
14 Harmonic Maps and Hodge Theory
359
19
14.1 The Eells-Sampson Theory
359
4
14.2 Harmonic and Pluriharmonic Maps
363
2
14.3 Applications to Locally Symmetric Spaces
365
9
14.4 Harmonic and Higgs Bundles
374
4
Appendices
378
37
A Projective Varieties and Complex Manifolds
378
5
B Homology and Cohomology
383
12
B.1 Simplicial Theory
383
4
B.2 Singular Theory
387
4
B.3 Manifolds
391
4
C Vector Bundles and Chern Classes
395
20
C.1 Vector Bundles
395
8
C.2 Axiomatic Introduction of Chern Classes
403
4
C.3 Connections, Curvature, and Chern Classes
407
5
C.4 Flat Connections
412
3
Bibliography
415
12
Subject Index
427