This is the English translation of Professor Voisin's book reflecting the discovery of the mirror symmetry phenomenon. The first chapter is devoted to the geometry of Calabi-Yau manifolds, and the second describes, as motivation, the ideas from quantum field theory that led to the discovery of mirror symmetry.

The other chapters deal with more specialized aspects of the subject: the work of Candelas, de la Ossa, Greene, and Parkes, based on the fact that under the mirror symmetry hypothesis, the variation of Hodge structure of a Calabi-Yau threefold determines the Gromov-Witten invariants of its mirror; Batyrev's construction, which exhibits the mirror symmetry phenomenon between hypersurfaces of toric Fano varieties, after a combinatorial classification of the latter; the mathematical construction of the Gromov-Witten potential, and the proof of its crucial property (that it satisfies the WDVV equation), which makes it possible to construct a flat connection underlying a variation of Hodge structure in the Calabi-Yau case. The book concludes with the first "naive" Givental computation, which is a mysterious mathematical justification of the computation of Candelas, et al.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students and research mathematicians interested in algebraic geometry; mathematical physicists.

Reviews

"This book ... might yet give, even to the non-specialist, some basic orientation in the complicated and rapidly developing world of mirror symmetry."

-- European Mathematical Society Newsletter

"Without any doubt, the English version of this panoramic introduction to the phenomenon of mirror symmetry will find a much larger number of interesting readers than the French original could do, and that is what this beautiful text really deserves."

-- Zentralblatt MATH

Table of Contents

• Calabi-Yau manifolds
• "Physical" origin of the conjecture
• The work of Candelas-de la Ossa-Green-Parkes
• The work of Batyrev
• Quantum cohomology
• The Givental construction
• Bibliography