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El. knyga: Period Mappings and Period Domains

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(University of Utah), (Johannes Gutenberg Universität Mainz, Germany),
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This up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. Starting with an explanation of Griffiths' basic theory, the authors go on to introduce spectral sequences and Koszul complexes that are used to derive results about cycles on higher-dimensional algebraic varieties such as the NoetherLefschetz theorem and Nori's theorem. They explain differential geometric methods, leading up to proofs of Arakelov-type theorems, the theorem of the fixed part and the rigidity theorem. They also use Higgs bundles and harmonic maps to prove the striking result that not all compact quotients of period domains are Kähler. This thoroughly revised second edition includes a new third part covering important recent developments, in which the group-theoretic approach to Hodge structures is explained, leading to MumfordTate groups and their associated domains, the MumfordTate varieties and generalizations of Shimura varieties.

Recenzijos

Review of previous edition: 'This book, dedicated to Philip Griffiths, provides an excellent introduction to the study of periods of algebraic integrals and their applications to complex algebraic geometry. In addition to the clarity of the presentation and the wealth of information, this book also contains numerous problems which render it ideal for use in a graduate course in Hodge theory.' Mathematical Reviews Review of previous edition: ' generally more informal and differential-geometric in its approach, which will appeal to many readers the book is a useful introduction to Carlos Simpson's deep analysis of the fundamental groups of compact Kähler manifolds using harmonic maps and Higgs bundles.' Burt Totaro, University of Cambridge 'This monograph provides an excellent introduction to Hodge theory and its applications to complex algebraic geometry.' Gregory Pearlstein, Nieuw Archief voor Weskunde

Daugiau informacijos

An introduction to Griffiths' theory of period maps and domains, focused on algebraic, group-theoretic and differential geometric aspects.
Preface to the Second Edition ix
Preface to the First Edition xi
PART ONE BASIC THEORY
1(186)
1 Introductory Examples
3(68)
1.1 Elliptic Curves
3(24)
1.2 Riemann Surfaces of Higher Genus
27(20)
1.3 Double Planes
47(15)
1.4 Mixed Hodge Theory Revisited
62(9)
2 Cohomology of Compact Kahler Manifolds
71(23)
2.1 Cohomology of Compact Differentiable Manifolds
71(6)
2.2 What Happens on Kahler Manifolds
77(11)
2.3 How Lefschetz Further Decomposes Cohomology
88(6)
3 Holomorphic Invariants and Cohomology
94(36)
3.1 Is the Hodge Decomposition Holomorphic?
94(8)
3.2 A Case Study: Hypersurfaces
102(10)
3.3 How Log-Poles Lead to Mixed Hodge Structures
112(5)
3.4 Algebraic Cycles and Their Cohomology Classes
117(5)
3.5 Tori Associated with Cohomology
122(4)
3.6 Abel--Jacobi Maps
126(4)
4 Cohomology of Manifolds Varying in a Family
130(30)
4.1 Smooth Families and Monodromy
130(3)
4.2 An Example: Lefschetz Fibrations and Their Topology
133(4)
4.3 Variations of Hodge Structures Make Their First Appearance
137(3)
4.4 Period Domains Are Homogeneous
140(8)
4.5 Period Maps
148(7)
4.6 Abstract Variations of Hodge Structure
155(2)
4.7 The Abel--Jacobi Map Revisited
157(3)
5 Period Maps Looked at Infinitesimally
160(27)
5.1 Deformations of Compact Complex Manifolds
160(4)
5.2 Enter: the Thick Point
164(3)
5.3 The Derivative of the Period Map
167(3)
5.4 An Example: Deformations of Hypersurfaces
170(4)
5.5 Infinitesimal Variations of Hodge Structure
174(3)
5.6 Application: A Criterion for the Period Map to be an Immersion
177(1)
5.7 Counterexamples to Infinitesimal Torelli
178(9)
PART TWO ALGEBRAIC METHODS
187(120)
6 Spectral Sequences
189(18)
6.1 Fundamental Notions
189(3)
6.2 Hypercohomology Revisited
192(4)
6.3 The Hodge Filtration Revisited
196(3)
6.4 Derived Functors
199(4)
6.5 Algebraic Interpretation of the Gauss--Manin Connection
203(4)
7 Koszul Complexes and Some Applications
207(23)
7.1 The Basic Koszul Complexes
207(3)
7.2 Koszul Complexes of Sheaves on Projective Space
210(3)
7.3 Castelnuovo's Regularity Theorem
213(6)
7.4 Macaulay's Theorem and Donagi's Symmetrizer Lemma
219(4)
7.5 Applications: The Noether--Lefschetz Theorems
223(7)
8 Torelli Theorems
230(25)
8.1 Infinitesimal Torelli Theorems
230(5)
8.2 Global Torelli Problems
235(5)
8.3 Generic Torelli for Hypersurfaces
240(5)
8.4 Moduli
245(10)
9 Normal Functions and Their Applications
255(17)
9.1 Normal Functions and Infinitesimal Invariants
255(7)
9.2 The Griffiths Group of Hypersurface Sections
262(5)
9.3 The Theorem of Green and Voisin
267(5)
10 Applications to Algebraic Cycles: Nori's Theorem
272(35)
10.1 A Detour into Deligne Cohomology with Applications
272(4)
10.2 The Statement of Nori's Theorem
276(5)
10.3 A Local-to-Global Principle
281(3)
10.4 Jacobi Modules and Koszul Cohomology
284(3)
10.5 Linking the Two Spectral Sequences Through Duality
287(3)
10.6 A Proof of Nori's Theorem
290(6)
10.7 Applications of Nori's Theorem
296(11)
PART THREE DIFFERENTIAL GEOMETRIC METHODS
307(96)
11 Further Differential Geometric Tools
309(19)
11.1 Chern Connections and Applications
309(4)
11.2 Subbundles and Quotient Bundles
313(4)
11.3 Principal Bundles and Connections
317(4)
11.4 Connections on Associated Vector Bundles
321(3)
11.5 Totally Geodesic Submanifolds
324(4)
12 Structure of Period Domains
328(18)
12.1 Homogeneous Bundles on Homogeneous Spaces
328(2)
12.2 Reductive Domains and Their Tangent Bundle
330(2)
12.3 Canonical Connections on Reductive Spaces
332(2)
12.4 Higgs Principal Bundles
334(3)
12.5 The Horizontal and Vertical Tangent Bundles
337(4)
12.6 On Lie Groups Defining Period Domains
341(5)
13 Curvature Estimates and Applications
346(37)
13.1 Higgs Bundles, Hodge Bundles, and their Curvature
347(9)
13.2 Logarithmic Higgs Bundles
356(3)
13.3 Polarized Variations Give Polystable Higgs Bundles
359(5)
13.4 Curvature Bounds over Curves
364(5)
13.5 Geometric Applications of Higgs Bundles
369(3)
13.6 Curvature of Period Domains
372(3)
13.7 Applications
375(8)
14 Harmonic Maps and Hodge Theory
383(20)
14.1 The Eells--Sampson Theory
383(4)
14.2 Harmonic and Pluriharmonic Maps
387(2)
14.3 Applications to Locally Symmetric Spaces
389(9)
14.4 Harmonic and Higgs Bundles
398(5)
PART FOUR ADDITIONAL TOPICS
403(84)
15 Hodge Structures and Algebraic Groups
405(23)
15.1 Hodge Structures Revisited
405(8)
15.2 Mumford--Tate Groups
413(6)
15.3 Mumford--Tate Subdomains and Period Maps
419(9)
16 Mumford--Tate Domains
428(25)
16.1 Shimura Domains
428(6)
16.2 Mumford--Tate Domains
434(5)
16.3 Mumford--Tate Varieties and Shimura Varieties
439(8)
16.4 Examples of Mumford--Tate Domains
447(6)
17 Hodge Loci and Special Subvarieties
453(34)
17.1 Hodge Loci
454(4)
17.2 Equivariant Maps Between Mumford--Tate Domains
458(4)
17.3 The Moduli Space of Cubic Surfaces is a Shimura Variety
462(9)
17.4 Shimura Curves and Their Embeddings
471(8)
17.5 Characterizations of Special Subvarieties
479(8)
Appendix A Projective Varieties and Complex Manifolds 487(5)
Appendix B Homology and Cohomology 492(12)
Appendix C Vector Bundles and Chern Classes 504(20)
Appendix D Lie Groups and Algebraic Groups 524(16)
References 540(16)
Index 556
James Carlson is Professor Emeritus at the University of Utah. From 2003 to 2012, he was president of the Clay Mathematics Institute, New Hampshire. Most of Carlson's research is in the area of Hodge theory. Stefan Müller-Stach is Professor of number theory at Johannes Gutenberg Universität Mainz, Germany. He works in arithmetic and algebraic geometry, focussing on algebraic cycles and Hodge theory, and his recent research interests include period integrals and the history and foundations of mathematics. Recently, he has published monographs on number theory (with J. Piontkowski) and period numbers (with A. Huber), as well as an edition of some works of Richard Dedekind. Chris Peters is a retired professor from the Université Grenoble Alpes, France and has a research position at the Eindhoven University of Technology, The Netherlands. He is widely known for the monographs Compact Complex Surfaces (with W. Barth, K. Hulek and A. van de Ven, 1984), as well as Mixed Hodge Structures, (with J. Steenbrink, 2008). He has also written shorter treatises on the motivic aspects of Hodge theory, on motives (with J. P. Murre and J. Nagel) and on applications of Hodge theory in mirror symmetry (with Bertin).
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