Period Mappings and Period Domains (Cambridge Studies in Advanced Mathematics) Book

The concept of a period of an integral goes back to the 19th century;it has been introduced by Legendre and Weierstra^D^ss for integrals ofcertain elliptic functions over certain closed circuits in thedissected complex plane. Rephrased in modern terminology, theseintegrals encode how the complex structure of an elliptic curvevaries. Griffiths generalized this idea and introduced period maps andperiod domains which reflect how the complex structure for higherdimensional varieties varies. These concepts and their basicproperties are discussed in the first part of the book. The algebraicaspects of this construction are investigated in the second part andbring the reader to the forefronts of research in cycletheory. In the third part differential geometrical aspects are studiedin detail and lead to another frontier of research in complexgeometry: Higgs bundles and harmonic theory.Read More