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Since its introduction, Hodge theory and its variants (variations of Hodge structures, mixed Hodge structures, variations of mixed Hodge structures) have been a source of some of the deepest results in algebraic geoemetry. The Hodge index theorem, for example, was one of the first results relating analytic invariants (the dimensions $h^{p,q}$ of the space of harmonic $(p,q)$-forms) with algebraic-topological invariants of an algebraic variety $X$ over~$\bC$; in a sense this is even a model for the much deeper Donaldson theory of four-manifolds. More recently the theory of variations of Hodge structures has served as the basis of the theory of Shimura varieties, and, similarly, mixed Hodge structures form a basis for a theory of mixed Shimura varieties, important for the theory of compactifications.

The book under review is a collection of three articles about Hodge theory and related developments, which are all aimed at non-experts and fulfill, in an extremely satisfactory manner, two functions. First, the basic methods used in the theories are discussed and developed in great detail; second, some newer developments are described, giving the reader a good overview of the more important applications. Furthermore, the style makes these articles a joy to work through, even for the mathematician not encountering these subjects for the first time. I now sketch the contents of the individual articles.

The first, by Demailly, concerns the $L^2$-theory of Hodge structures. Although Hodge theory was not originally introduced in this manner, the method is very appropriate today, especially concerning applications to non-compact or singular varieties. The $L^2$-condition is just right to carry over basic results to this context. This paper consists of two parts. In the first the general theory is derived. The author is very thorough in the development, and in particular, the entire theory of pseudo-differential elliptic operators (necessary for the finite-dimensionality of Hodge groups) is developed in the more general context. Similarly, the theory of K\"ahler manifolds is presented in this context. One section is devoted to the Hodge-Fr\"ohlicher spectral sequence, which describes exactly how, for a general (not necessarily K\"ahler) compact complex manifold, the actual Hodge groups deviate from the accustomed (K\"ahler) symmetry $H^{p,q}=\overline{H^{q,p}}$.

In the second part of this paper, vanishing results and applications are discussed, again everything in the $L^2$-context. A very complete explanation of the notions of pseudoconvexity and positivity of vector bundles is given, together with the corresponding vanishing results. The Bochner method is described in detail, with several applications. It turns out that the $L^2$-methods allow a simplified proof of some very recent results of Y.~T. Siu [Invent.\ Math.\ {\bf 124} (1996), no. 1-3, 563--571; MR1369428 (97a:32036) culminating in an effective version of Matsusaka's big theorem (an effective bound $m_0$ so that, for positive $L$ and all $m\geq m_0$, $L^m$ is very ample).

The second article, by Illusie, is concerned with a very different aspect: applications of characteristic $p>0$ methods to characteristic $0$, in particular, the proof of degeneration of the Hodge spectral sequence and the vanishing theorem of Kodaira-Akizuki-Nakano of P. Deligne and Illusie [Invent.\ Math.\ {\bf 89} (1987), no. 2, 247--270; MR0894379 (88j:14029)] which applied these methods. Again the level of presentation is for non-specialists and contains a fair amount of rather inaccessible background material, written in a very lucid and understandable manner. One section discusses the notions of smoothness and liftability (essentially EGA material). The next introduces and describes in detail the principal characteristic $p$ methods: Frobenius and the Cartier isomorphism. The following section sketches the homological algebra needed to formulate the result: derived categories and spectral sequences. After the proof of the main theorem of Deligne and Illusie [op. cit.] the "well-known" methods used to deduce from characteristic $p$ the corresponding statements in characteristic $0$ is described in detail. An additional section considers more recent results and open problems.

Finally, the third article, by Bertin and Peters, discusses variations of Hodge structures and its applications to mirror symmetry. This article also consists of two parts, the first with general theory, the second with more specific applications in mind. In particular, the discussion of the Gauss-Manin connection is done in great detail, as this is one of the fundamentals of the entire theory and in particular of the applications which are to follow. The local monodromy theorem is described, and a proof, due to W. Schmid, is sketched; also, other results of the fundamental work of his [Invent.\ Math.\ {\bf 22} (1973), 211--319; MR0382272 (52 \#3157)] are described, like the limit mixed Hodge structure. Further topics include the local invariant cycle theorem and the status of vanishing cycles. This leads naturally to the complicated theory of Hodge modules due to Sato, which is then introduced. In the second part, after a brief introduction to mirror symmetry, the Picard-Fuchs equations of a family of Calabi-Yau manifolds is derived in great detail. In the last section, the authors then give a new interpretation of the mirror symmetry conjecture:

For every $q\in \Delta^*$, the mixed structure on $H^+(M^*)\times \{q\}$ coincides with the mixed structure of Deligne on $H^3(M_q)$. (Notation: $M^*$ is the generic member of the mirror family and $H^+$ denotes the even cohomology.)

Reviewed by Bruce Hunt 

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