There are many texts concerning the aspects of mirror symmetry having to do with variations of Hodge structure and counting curves, but only difficult research articles about the more recent geometry of mirror symmetry having to do with D-branes, homological mirror symmetry and torus fibrations. A reasonably self-contained and introductory book on the latter is therefore welcome and timely. Similarly there are introductory texts on the basics of hyper-Kähler geometry, but most of the deep results about compact hyper-Kähler manifolds are contained only in research papers. This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kähler manifolds, exceptional holonomy and mirror symmetry. In some sense hyper-Kähler manifolds can be thought of as a particularly nice subset of Calabi-Yau manifolds, and have a mirror symmetry of their own, so the choice of topics is a sensible one.

The first section is Joyce's contribution. It begins with a rapid survey of differential and Riemannian geometry: connections, curvature, holonomy, calibrations, and even symplectic geometry and moment maps. It is not the place to learn the material for the first time, but it is a good reference. There follows a survey and review of the SYZ conjecture in mirror symmetry [A. Strominger, S.-T. Yau and E. Zaslow, Nuclear Phys. B 479 (1996), no. 1-2, 243--259; MR1429831 (97j:32022)] and the author's many constructions of local special Lagrangian singularities and fibrations that are so important in our current understanding of the conjecture. This is extremely useful as a reference, as until now the results have only been available in hundreds of pages of technical preprints. The main ideas are successfully conveyed without including complete proofs, and their impact on the SYZ conjecture (such as different codimension one discriminant loci for a fibration and its dual) is explained. (Extensions of) McLean's deformation theory [R. C. McLean, Comm. Anal. Geom. 6 (1998), no. 4, 705--747; MR1664890 (99j:53083)] for calibrated submanifolds (and especially special Lagrangians) are also described.

Part II, by Gross, describes Calabi-Yau manifolds from a more classical algebro-geometric point of view, and mirror symmetry both old and new, with the two strands nicely tied together. Kodaira-Spencer deformation theory and its unobstructedness for Calabi-Yaus (the Bogomolov-Tian-Todorov theorem) is reviewed, and a lightning tour is given of Batyrev's mirror construction, Gromov-Witten theory and the variation and degeneration of Hodge structures. This is probably the hardest part of the book and is not so self-contained (Schmid's work on Hodge structures, the nilpotent orbit theorem, Morrison's definition of large complex structure limit points), but ties in well with later material of the author's which sheds light on and simplifies these concepts using torus fibrations. The Greene-Plesser mirror of the quintic is described, and a good deal of the seminal calculation [P. Candelas et al., Nuclear Phys. B 359 (1991), no. 1, 21--74; MR1115626 (93b:32029)] of periods on the mirror quintic to derive the mirror map and the prediction of numbers of rational curves on the quintic. Finally Gross turns to the SYZ conjecture, Lagrangian torus fibrations and their duals, and integral affine structures (`action-angle coordinates') on the bases of these fibrations. Via his work (see for instance ``Special Lagrangian fibrations. I. Topology'' [in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 156--193, World Sci. Publishing, River Edge, NJ, 1998; MR1672120 (2000e:14066)]) this beautifully explains the flipping of Hodge numbers between mirror manifolds, large complex and Kähler limit points, and monodromy around them. This is nicely explained for smooth fibrations.

The final chapter, by Huybrechts, is devoted to hyper-Kähler manifolds (although some of the chapter applies to more general holomorphic symplectic manifolds). In many ways the moral is that they are much more like K3 surfaces than Calabi-Yau 3-folds. Many of the powerful tools for studying K3 surfaces extend, at least partially, to general hyper-Kähler manifolds, possibly after allowing for birational modifications. The main tools are explained in detail, in particular deformation theory, period mappings, local Torelli, the remarkable Beauville-Bogomolov pairing, twistor space and hyper-Kähler rotation, and Mukai flops. Some of the surprising consequences are proved, such as Matsushita's theorem on fibrations of hyper-Kähler manifolds [D. Matsushita, Topology 38 (1999), no. 1, 79--83; MR1644091 (99f:14054)], and the second author's deep work on the structure of the Kähler cone and deformation equivalence of birational hyper-Kähler manifolds [Math. Ann. 326 (2003), no. 3, 499--513; MR1992275 ].

This is an excellent and useful book. The different chapters mostly fit together reasonably well (with the possible exception of the Tian-Todorov theorem whose motivation is described beautifully by Huybrechts in Chapter 3, after the rather dry technical proof in Chapter 2). It will be popular with anyone trying to learn about Kähler manifolds with c₁=0 and mirror symmetry.

Reviewed by Richard Thomas. Back