**SMF/AMS Texts and Monographs**

Volume: 1;
1999;
120 pp;
Softcover

MSC: Primary 14;
Secondary 32; 81; 53

Print ISBN: 978-0-8218-1947-0

Product Code: SMFAMS/1

List Price: $32.00

AMS Member Price: $25.60

MAA Member Price: $28.80

**Electronic ISBN: 978-1-4704-3579-0
Product Code: SMFAMS/1.E**

List Price: $32.00

AMS Member Price: $25.60

MAA Member Price: $28.80

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# Mirror Symmetry

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*Claire Voisin*

A co-publication of the AMS and Société Mathématique de France

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 & Endorsements

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