## 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.)

### Citations:

Zbl 0114.39402; Zbl 0853.57030
Full Text: