Period mappings and period domains /
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 tha...
| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge :
Cambridge University Press,
[2017]
Cambridge, United Kingdom : 2017 Cambridge, United Kingdom ; New York, NY : 2017 |
| Edition: | Second edition |
| Series: | Cambridge studies in advanced mathematics ;
168 Cambridge studies in advanced mathematics ; 168 |
| Subjects: |
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| 100 | 1 | |a Carlson, James A., |d 1946- |e author |0 http://viaf.org/viaf/46835095 | |
| 100 | 1 | |a Carlson, James A., |d 1946- |e author |1 http://viaf.org/viaf/46835095 | |
| 100 | 1 | |a Carlson, James A., |d 1946- |e author |= ^A1630173 | |
| 100 | 1 | |a Carlson, James A., |d 1946- |e author | |
| 100 | 1 | |a Carlson, James A., |d 1946- | |
| 245 | 1 | 0 | |a Period mappings and period domains / |c James Carlson (University of Utah), Stefan Müller-Stach (Johannes Gutenberg Universität, Mainz, Germany), Chris Peters (Université Grenoble Alpes, France ) |
| 250 | |a Second edition | ||
| 263 | |a 1708 | ||
| 264 | 1 | |a Cambridge : |b Cambridge University Press, |c [2017] | |
| 264 | 1 | |a Cambridge, United Kingdom : |b Cambridge University Press, |c 2017 | |
| 264 | 1 | |a Cambridge, United Kingdom ; |a New York, NY : |b Cambridge University Press, |c 2017 | |
| 264 | 4 | |c ©2017 | |
| 300 | |a pages cm | ||
| 300 | |a xiv, 562 pages : |b illustrations ; |c 23 cm | ||
| 300 | |a xiv, 562 pages : |b illustrations ; |c 24 cm | ||
| 300 | |a xiv, 562 pages ; |c 23 cm | ||
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| 338 | |a volume |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics ; |v 168 | |
| 500 | |a Previous edition: 2003 | ||
| 504 | |a Includes bibliographical references (pages 540-555) and index | ||
| 504 | |a Includes bibliographical references and index | ||
| 505 | 0 | |a Part one. Basic theory -- 1. Introductory examples -- 2. Cohomology of compact Kähler manifolds -- 3. Holomorphic invariants and cohomology -- 4. Cohomology of manifolds varying in a family -- 5. Period maps looked at infinitesimally -- Part two. Algebraic methods -- 6. Spectral sequences -- 7. Koszul complexes and some applications -- 8. Torelli theorems -- 9. Normal functions and their applications -- 10. Applications to algebraic cycles : Nori's theorem -- Part three. Differential geometric methods -- 11. Further differential geometric tools -- 12. Structure of period domains -- 13. Curvature estimates and applications -- 14. Harmonic maps and Hodge theory -- Part four. Additional topics -- 15. Hodge structures and algebraic groups -- 16. Mumford-Tate domains -- 17. Hodge loci and special subvarieties -- Appendix A. Projective varieties and complex manifolds -- Appendix B. Homology and cohomology -- Appendix C. Vector bundles and Chern classes -- Appendix D. Lie groups and algebraic groups | |
| 505 | 0 | 0 | |a note: |g pt. ONE |t BASIC THEORY -- |g 1 |t Introductory Examples -- |g 1.1. |t Elliptic Curves -- |g 1.2. |t Riemann Surfaces of Higher Genus -- |g 1.3. |t Double Planes -- |g 1.4. |t Mixed Hodge Theory Revisited -- |g 2. |t Cohomology of Compact Kahler Manifolds -- |g 2.1. |t Cohomology of Compact Differentiable Manifolds -- |g 2.2. |t What Happens on Kahler Manifolds -- |g 2.3. |t How Lefschetz Further Decomposes Cohomology -- |g 3. |t Holomorphic Invariants and Cohomology -- |g 3.1. |t Is the Hodge Decomposition Holomorphic? -- |g 3.2. |t Case Study: Hypersurfaces -- |g 3.3. |t How Log-Poles Lead to Mixed Hodge Structures -- |g 3.4. |t Algebraic Cycles and Their Cohomology Classes -- |g 3.5. |t Tori Associated with Cohomology -- |g 3.6. |t Abel--Jacobi Maps -- |g 4. |t Cohomology of Manifolds Varying in a Family -- |g 4.1. |t Smooth Families and Monodromy -- |g 4.2. |t Example: Lefschetz Fibrations and Their Topology -- |g 4.3. |t Variations of Hodge Structures Make Their First Appearance -- |g 4.4. |t Period Domains Are Homogeneous -- |g 4.5. |t Period Maps -- |g 4.6. |t Abstract Variations of Hodge Structure -- |g 4.7. |t Abel--Jacobi Map Revisited -- |g 5. |t Period Maps Looked at Infinitesimally -- |g 5.1. |t Deformations of Compact Complex Manifolds -- |g 5.2. |t Enter: the Thick Point -- |g 5.3. |t Derivative of the Period Map -- |g 5.4. |t Example: Deformations of Hypersurfaces -- |g 5.5. |t Infinitesimal Variations of Hodge Structure -- |g 5.6. |t Application: A Criterion for the Period Map to be an Immersion -- |g 5.7. |t Counterexamples to Infinitesimal Torelli -- |g pt. TWO |t ALGEBRAIC METHODS -- |g 6. |t Spectral Sequences -- |g 6.1. |t Fundamental Notions -- |g 6.2. |t Hypercohomology Revisited -- |g 6.3. |t Hodge Filtration Revisited -- |g 6.4. |t Derived Functors -- |g 6.5. |t Algebraic Interpretation of the Gauss--Manin Connection -- |g 7. |t Koszul Complexes and Some Applications -- |g 7.1. |t Basic Koszul Complexes -- |g 7.2. |t Koszul Complexes of Sheaves on Projective Space -- |g 7.3. |t Castelnuovo's Regularity Theorem -- |g 7.4. |t Macaulay's Theorem and Donagi's Symmetrizer Lemma -- |g 7.5. |t Applications: The Noether--Lefschetz Theorems -- |g 8. |t Torelli Theorems -- |g 8.1. |t Infinitesimal Torelli Theorems -- |g 8.2. |t Global Torelli Problems -- |g 8.3. |t Generic Torelli for Hypersurfaces -- |g 8.4. |t Moduli -- |g 9. |t Normal Functions and Their Applications -- |g 9.1. |t Normal Functions and Infinitesimal Invariants -- |g 9.2. |t Griffiths Group of Hypersurface Sections -- |g 9.3. |t Theorem of Green and Voisin -- |g 10. |t Applications to Algebraic Cycles: Nori's Theorem -- |g 10.1. |t Detour into Deligne Cohomology with Applications -- |g 10.2. |t Statement of Nori's Theorem -- |g 10.3. |t Local-to-Global Principle -- |g 10.4. |t Jacobi Modules and Koszul Cohomology -- |g 10.5. |t Linking the Two Spectral Sequences Through Duality -- |g 10.6. |t Proof of Nori's Theorem -- |g 10.7. |t Applications of Nori's Theorem -- |g pt. THREE |t DIFFERENTIAL GEOMETRIC METHODS -- |g 11. |t Further Differential Geometric Tools -- |g 11.1. |t Chern Connections and Applications -- |g 11.2. |t Subbundles and Quotient Bundles -- |g 11.3. |t Principal Bundles and Connections -- |g 11.4. |t Connections on Associated Vector Bundles -- |g 11.5. |t Totally Geodesic Submanifolds -- |g 12. |t Structure of Period Domains -- |g 12.1. |t Homogeneous Bundles on Homogeneous Spaces -- |g 12.2. |t Reductive Domains and Their Tangent Bundle -- |g 12.3. |t Canonical Connections on Reductive Spaces -- |g 12.4. |t Higgs Principal Bundles -- |g 12.5. |t Horizontal and Vertical Tangent Bundles -- |g 12.6. |t On Lie Groups Defining Period Domains -- |g 13. |t Curvature Estimates and Applications -- |g 13.1. |t Higgs Bundles, Hodge Bundles, and their Curvature -- |g 13.2. |t Logarithmic Higgs Bundles -- |g 13.3. |t Polarized Variations Give Polystable Higgs Bundles -- |g 13.4. |t Curvature Bounds over Curves -- |g 13.5. |t Geometric Applications of Higgs Bundles -- |g 13.6. |t Curvature of Period Domains -- |g 13.7. |t Applications -- |g 14. |t Harmonic Maps and Hodge Theory -- |g 14.1. |t Eells--Sampson Theory -- |g 14.2. |t Harmonic and Pluriharmonic Maps -- |g 14.3. |t Applications to Locally Symmetric Spaces -- |g 14.4. |t Harmonic and Higgs Bundles -- |g pt. FOUR |t ADDITIONAL TOPICS -- |g 15. |t Hodge Structures and Algebraic Groups -- |g 15.1. |t Hodge Structures Revisited -- |g 15.2. |t Mumford--Tate Groups -- |g 15.3. |t Mumford--Tate Subdomains and Period Maps -- |g 16. |t Mumford--Tate Domains -- |g 16.1. |t Shimura Domains -- |g 16.2. |t Mumford--Tate Domains -- |g 16.3. |t Mumford--Tate Varieties and Shimura Varieties -- |g 16.4. |t Examples of Mumford--Tate Domains -- |g 17. |t Hodge Loci and Special Subvarieties -- |g 17.1. |t Hodge Loci -- |g 17.2. |t Equivariant Maps Between Mumford--Tate Domains -- |g 17.3. |t Moduli Space of Cubic Surfaces is a Shimura Variety -- |g 17.4. |t Shimura Curves and Their Embeddings -- |g 17.5. |t Characterizations of Special Subvarieties. |
| 520 | |a 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 Noether{u2013}Lefschetz 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 Mumford{u2013}Tate groups and their associated domains, the Mumford{u2013}Tate varieties and generalizations of Shimura varieties.--Provided by publisher | ||
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