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Cambridge Studies in Advanced Mathematics Ser.: Period Mappings and Period Domains by Stefan Müller-Stach, James Carlson and Chris Peters (2003, Hardcover)
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About this product
Product Identifiers
PublisherCambridge University Press
ISBN-100521814669
ISBN-139780521814669
eBay Product ID (ePID)2320493
Product Key Features
Number of Pages448 Pages
LanguageEnglish
Publication NamePeriod Mappings and Period Domains
SubjectTopology, Geometry / Algebraic
Publication Year2003
TypeTextbook
AuthorStefan Müller-Stach, James Carlson, Chris Peters
Subject AreaMathematics
SeriesCambridge Studies in Advanced Mathematics Ser.
FormatHardcover
Dimensions
Item Height1.1 in
Item Weight26 Oz
Item Length9.3 in
Item Width6.3 in
Additional Product Features
Intended AudienceScholarly & Professional
LCCN2003-043474
Dewey Edition21
Reviews"[T]he work of Carlson, Müller-Stach, and Peters gives structure and overview to this important but sometimes relatively inaccessible branch of geometry." SIAM Review, 'The presentation of the vast material is very lucid and inspiring, methodologically well-planned and utmost user-friendly considering such sophisticated a complex of topics.' Zentralblatt fr Mathematik, '… 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 Khler manifolds using harmonic maps and Higgs bundles.' Burt Totaro, University of Cambridge, 'The presentation of the vast material is very lucid and inspiring, methodologically well-planned and utmost user-friendly considering such sophisticated a complex of topics.' Zentralblatt für Mathematik, '... 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 book, dedicated to Phillip 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." Gregory J. Pearlstein, Mathematical Reviews, '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, "[T]he work of Carlson, M&üller-Stach, and Peters gives structure and overview to this important but sometimes relatively inaccessible branch of geometry." SIAM Review
Series Volume NumberSeries Number 85
IllustratedYes
Dewey Decimal516.3/5
Table Of ContentPart 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.
SynopsisThe 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., The concept of a period of an elliptic integral goes back to the 18th century. Later Abel, Gauss, Jacobi, Legendre, Weierstrass and others made a systematic study of these integrals. Rephrased in modern terminology, these give a way to encode how the complex structure of a two-torus varies, thereby showing that certain families contain all elliptic curves. Generalizing to higher dimensions resulted in the formulation of the celebrated Hodge conjecture, and in an attempt to solve this, Griffiths generalized the classical notion of period matrix and introduced period maps and period domains which reflect how the complex structure for higher dimensional varieties varies. The basic theory as developed by Griffiths is explained in the first part of the book. Then, in the second part spectral sequences and Koszul complexes are introduced and are used to derive results about cycles on higher dimensional algebraic varieties such as the Noether-Lefschetz theorem and Nori's theorem. Finally, in the third part differential geometric methods are explained leading up to proofs of Arakelov-type theorems, the theorem of the fixed part, the rigidity theorem, and more. Higgs bundles and relations to harmonic maps are discussed, and this leads to striking results such as the fact that compact quotients of certain period domains can never admit a Kahler metric or that certain lattices in classical Lie groups can't occur as the fundamental group of a Kahler manifold., 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.
LC Classification NumberQA564 .C28 2003
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