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. Structurally, the book is divided into three distinct parts, which are organized as follows.

Part I begins with several illustrative examples from the Hodge theory of elliptic curves and Riemann surfaces. Building on these examples, the authors then discuss the Hodge theory of higher dimensional varieties, culminating in Griffiths definition of the period map and P. Deligne's theory of mixed Hodge structures via log-poles.

Part II is devoted to spectral sequences and other algebraic techniques. The main applications considered here are the various Torelli theorems, normal functions, and the proof of Nori's theorem.

Part III considers the period map from the standpoint of differential geometry using the theory of harmonic maps and the curvature properties of the Hodge bundles. The main results are the rigidity of variations of Hodge structure, the Eells-Sampson theorem, Siu's Rigidity theorem, Higgs bundles, and applications to Kähler groups.

Reviewed by Gregory J. Pearlstein