Period Mappings and Period Domains
Period Mappings and Period Domains
Stefan Müller-Stach, Johannes Gutenberg Universität Mainz, Germany
Chris Peters, Université Grenoble Alpes, France
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–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–Tate groups and their associated domains, the Mumford–Tate varieties and generalizations of Shimura varieties.
- A completely revised and up-to-date new edition, covering all major new developments in the field
- Accessible to graduate students with modest backgrounds in algebraic topology and algebra
- Begins by providing a comprehensive introduction to the basic theory as developed by Griffiths
Reviews & endorsements
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
Product details
August 2017Hardback
9781108422628
590 pages
238 × 160 × 42 mm
1.03kg
35 b/w illus. 3 tables 180 exercises
Available
Table of Contents
- Part I. 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 II. 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 III. Differential Geometric Aspects:
- 11. Further differential geometric tools
- 12. Structure of period domains
- 13. Curvature estimates and applications
- 14. Harmonic maps and Hodge theory
- Part IV. 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
- References
- Index.
