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Homological algebra of mirror symmetry. (English) Zbl 0846.53021

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120-139 (1995).
The author proposes a “homological mirror conjecture” relating mirror symmetry to general structures of homological algebra. Let \(V\) be a \(2n\)-dimensional symplectic manifold with \(c_1 (V) = 0\) and \(W\) be a dual \(n\)-dimensional complex algebraic manifold. Let \(LV\) be the space of pairs \((x,L)\), where \(x\) is a point of \(V\) and \(L\) is a Lagrangian subspace of \(T_x V\). There exists a \(\mathbb Z\)-covering \(\widetilde {LV}\) of \(LV\) inducing a universal cover of each fiber. K. Fukaya [Morse homotopy, \(A_\infty\)-category and Floer homologies, MSRI preprint No. 020-94 (1993), see also Kim, Hong-Jong (ed.), Proceedings of the GARC workshop on geometry and topology ’93 held at the Seoul National University, Seoul, Korea, July 1993. Seoul: Seoul National University, Lect. Notes Ser., Seoul. 18, 1–102 (1993; Zbl 0853.57030)], based on ideas of Donaldson, Floer and Segal, constructed an \(A_\infty\)-category \(F(V)\) having as objects the Lagrangian submanifolds \({\mathcal L} \subset V\) endowed with a continuous lift \({\mathcal L} \to \widetilde {LV}\) of the map \({\mathcal L} \to LV\). (An \(A_\infty\)-category \(C\) is a collection of objects and \(\mathbb Z\)-graded spaces of morphisms \(\operatorname{Hom}_C (X,Y)\) endowed with higher compositions of morphisms satisfying relations similar to the defining relations of \(A_\infty\)-algebras; an \(A_\infty\)-algebra is a concept introduced by J. D. Stasheff [Trans. Am. Math. Soc. 108, 275–292, 293–312 (1963; Zbl 0114.39402)].)
The conjecture says that the derived category \(D^b (F(V))\) (or a suitable enlarged one) is equivalent to the derived category \(D^b (\text{Coh} (W))\) of coherent sheaves on \(W\).
For the entire collection see [Zbl 0829.00014].

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G05 Deformations of complex structures
18E30 Derived categories, triangulated categories (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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