Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. Topics vary each year but there are always three series of lectures by invited experts with evening exercises. The school held in June 2001 was devoted to recent interaction between differential and algebraic geometry. The book consists of notes written by lecturers of the corresponding three series of lectures. The first contribution (by D. Joyce) is devoted to the introduction and study of Riemannian properties. The last contribution (by D. Huybrechts) describes compact hyperkähler manifolds. The class of compact hyperkähler manifolds form an interesting class of Ricci-flat manifolds, playing an important role in different branches of mathematics and mathematical physics. In this part, many interesting topics are presented (including theory of holomorphic symplectic manifolds, deformation theory of complex structures, cohomology and other properties of compact hyperkähler manifolds, twistor space and moduli space of a hyperkähler manifold and a discussion of the projectivity of hyperkähler metrics). The themes of all three contributions are interrelated and together they give a nice introduction into a very interesting field of research on the border between mathematics and physics. I would like to strongly recommend the book to anybody interested in the topic.

Reviewer:

jbu