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Mirror symmetry. Transl. from the French by Roger Cooke. (English) Zbl 0945.14021
SMF/AMS Texts and Monographs. 1. Providence, RI: American Mathematical Society (AMS). Paris: Société Mathématique de France, xix, 120 p. (1999).
This book is the English translation of the French original, which appeared in 1996 under the title “Symétrie miroir” in the series “Panoramas et synthèses” edited by the Société Mathématique de France (cf. Zbl 0849.14001). As mirror symmetry has grown into one of the central topics in both algebraic geometry and mathematical physics (quantum field theory), C. Voisin’s panoramic introduction to the subject (at its state of art in 1995) has maintained its basic value as a first primer for mathematicians and physicists who want to become acquainted with this fascinating phenomenon. Also, this booklet may be seen as a first step towards a deeper study of mirror symmetry and its related topics in geometry and physics, whether by means of the rapidly growing wealth of original papers or with the help of the very recent comprehensive monograph “Mirror symmetry and algebraic geometry” by D. A. Cox and S. Katz [Math. Surv. Monographs 68 (1999)].
In this English translation, the original text has been left completely intact. However, the author has enhanced the introduction by a “note added in translation”, in which she comments on some recent work on mirror symmetry. In fact, since the appearance of the French original in 1996, the subject of mirror symmetry has developed in several directions, especially with respect to a better understanding of Gromov-Witten invariants and their significance, and the most important contributions to these recent developments have been listed in an appendix to the original bibliography, together with a few brief hints to them added to the introduction.
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.

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J81 Relationships with physics
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14J45 Fano varieties
14J30 \(3\)-folds
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)