The book serves as an introduction to the theory of period mappings for variation of a Hodge structure. It contains three chapters. The first part is devoted to a basic overview of the subject, starting from classical results for period integrals of elliptic curves, Hodge structure on cohomology of complex manifolds, and holomorphic families of invariants related to cohomology. It culminates in the central concept of the book illustrated by the role of monodromy for Lefschetz pencils. The second part treats some technical details, which allow the authors to tie up loose ends from the first section. The chapter starts with the machinery of spectral sequences and goes on to Koszul complexes, normal functions and Nori's theorem as a tool to investigate algebraic cycles. The final chapter of the book turns to purely differential aspects of period domains. The main goal is an explanation of the curvature properties relevant for period maps.

Reviewer:

pso