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Homological geometry and mirror symmetry. (English) Zbl 0863.14021
Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 472-480 (1995).
Two compact Kähler manifolds \(X\) and \(Y\) of the same dimension are called geometric mirrors of each other if their tables of Hodge numbers are mirror-symmetric. The phenomenon of the existence of mirror symmetry in complex geometry, first predicted and discovered by physicists in the context of conformal quantum field theory (string models), has recently become a topic of great interest and significance in both complex algebraic geometry and mathematical physics.
The present article, written about three years ago, provides a concise survey on the state of knowledge of mirror symmetry (up to 1994) and its relations to various topics in algebraic geometry and topological conformal quantum field theory. In the meantime, most of the material sketched here can be found, in a systematic, detailed, self-contained and up-to-date form, in the recent monograph “Symétrie miroir” by C. Voisin (Panoramas et Synthèses, Vol. 2, (1996; Zbl 0849.14001), where also the author’s own contributions to the subject [see the author, Sel. Math., New Ser. 1, No. 2, 325-345 (1995) and the author and B. Kim, Commun. Math. Phys. 168, No. 3, 609-641 (1995; Zbl 0828.55004)] are thoroughly discussed within the general, brand-new theory of mirror manifolds.
For the entire collection see [Zbl 0829.00014].

14J30 \(3\)-folds
81T70 Quantization in field theory; cohomological methods
58Z05 Applications of global analysis to the sciences
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics