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Period Mappings and Period Domains - James Carlson, Stefan Müller-Stach, Chris Peters

Period Mappings and Period Domains

Buch | Hardcover
448 Seiten
2003
Cambridge University Press (Verlag)
978-0-521-81466-9 (ISBN)
CHF 189,95 inkl. MwSt
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Griffiths generalised the idea of a period of an integral and introduced period maps and period domains which reflect the complex structures for higher-dimensional varieties. This 2003 book discusses these concepts and their basic properties, the algebraic aspects and the differential geometrical aspects leading to Higgs bundles and harmonic theory.
The period matrix of a curve effectively describes how the complex structure varies; this is Torelli's theorem dating from the beginning of the nineteenth century. In the 1950s during the first revolution of algebraic geometry, attention shifted to higher dimensions and one of the guiding conjectures, the Hodge conjecture, got formulated. In the late 1960s and 1970s Griffiths, in an attempt to solve this conjecture, generalized the classical period matrices introducing period domains and period maps for higher-dimensional manifolds. He then found some unexpected new phenomena for cycles on higher-dimensional algebraic varieties, which were later made much more precise by Clemens, Voisin, Green and others. This 2003 book presents this development starting at the beginning: the elliptic curve. This and subsequent examples (curves of higher genus, double planes) are used to motivate the concepts that play a role in the rest of the book.

Part I. Basic Theory of the Period Map: 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. The Period Map: Algebraic Methods: 6. Spectral sequences; 7. Koszul complexes and some applications; 8. Further applications: Torelli theorems for hypersurfaces; 9. Normal functions and their applications; 10. Applications to algebraic cycles: Nori's theorem; Part III: Differential Geometric Methods: 11. Further differential geometric tools; 12. Structure of period domains; 13. Curvature estimates and applications; 14. Harmonic maps and Hodge theory; Appendix A. Projective varieties and complex manifolds; Appendix B. Homology and cohomology; Appendix C. Vector bundles and Chern classes.

Erscheint lt. Verlag 20.10.2003
Reihe/Serie Cambridge Studies in Advanced Mathematics
Verlagsort Cambridge
Sprache englisch
Maße 159 x 237 mm
Gewicht 737 g
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Geometrie / Topologie
ISBN-10 0-521-81466-9 / 0521814669
ISBN-13 978-0-521-81466-9 / 9780521814669
Zustand Neuware
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