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 c_{1}=0 and mirror symmetry.

**Reviewed by Richard Thomas.** Back