This is an English translation of the text, which was originally published in French by the Société Mathématique de France as vol. 3 of the series Panoramas et Synthèses in 1996. It consists of three parts: Hodge Theory and Vanishing Theorems (by J.-P. Demailly), Frobenius and Hodge Degeneration (by L. Illusie), and Variations of Hodge Structure, Calabi-Yau Manifolds and Mirror Symmetry (by J. Bertin and Ch. Peters). Its main aim is to make the reader familiar with the contemporary state and development of the Hodge theory. The authors were well aware of the fact that the text would require prerequisities from several branches of mathematics. Consequently, they prepared the exposition in such a way that it is as easy as possible for the reader. Their attempt was quite successful, nevertheless a potential reader must have a basic knowledge in several directions (e.g., analytic and algebraic geometry, sheaves, homological algebra). But it is worth the effort, because he (she) can penetrate into the contemporary state of affair in the Hodge theory, and it is not easy to find other texts having the same qualities. Moreover, some recent publications on this topic are written in the language of physics, which need not always be completely understandable to all mathematicians. Numerous references contribute to a good orientation in the subject. We find here also many open problems showing to young scientists possible ways of further research.

Reviewer:

jiva