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Mirror symmetry in dimension 3. (English) Zbl 0880.14019

Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 275-293, Exp. No. 801 (1996).
(From the abstract:) Mirror symmetry is a partial duality between Calabi-Yau manifolds, i.e., complex manifolds with \(c_1 =0\) and Ricci flat Kähler metrics. It was discovered in physics as an equivalence of string theories on different backgrounds. Conjecturally, the generating function for appropriately defined numbers of rational curves of all degrees on one Calabi-Yau manifold is related with the variation of Hodge structures over the universal family of complex structures on the dual manifold.
The case of complex dimension 3 is especially interesting for physics and mathematics. We review relevant results and constructions from algebraic geometry and symplectic topology. Also we describe holomorphic anomaly equations giving predictions for numbers of curves of positive genera.
For the entire collection see [Zbl 0851.00039].

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32J27 Compact Kähler manifolds: generalizations, classification
14D07 Variation of Hodge structures (algebro-geometric aspects)
81T50 Anomalies in quantum field theory
14J30 \(3\)-folds
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