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  "body": "\n## Statement\n\nLet $X$ be a compact Calabi–Yau manifold, viewed as a symplectic manifold.\nLet $\\check{X}$ be a Calabi–Yau manifold that is mirror to $X$ in the sense of mirror symmetry.\nThen the derived category of coherent sheaves $D^b\\mathrm{Coh}(\\check{X})$ is equivalent, as a triangulated category, to the split-closed derived [[Fukaya category|4HS60JABC5]] $D^\\pi \\mathcal{F}(X)$.\nThe same holds with the roles of $X$ and $\\check{X}$ exchanged.\n\n## Remarks\n\nKontsevich proposed the conjecture in his address to the International Congress of Mathematicians in Zürich in 1994, as a categorical explanation of mirror symmetry.\nThe conjecture is not a single precise statement, because \"mirror\" is not defined in general.\nIt is proved in many cases where a mirror pair is given by an explicit construction.\nPolishchuk and Zaslow proved it for elliptic curves, Seidel for the quartic surface, Abouzaid and Smith for the four-dimensional torus, and Sheridan for Calabi–Yau hypersurfaces in projective space.\nVersions for Fano manifolds, whose mirrors are Landau–Ginzburg models, replace the derived category on one side by a category of matrix factorizations or a Fukaya–Seidel category.\n\n## Explanations\n\n### L1 human\n\nMirror symmetry is a duality from string theory that pairs two very different spaces.\nOn one side one looks at the symplectic geometry of a space, through its Lagrangian submanifolds.\nOn the other side one looks at the algebraic geometry of the mirror space, through its coherent sheaves.\nHomological mirror symmetry predicts that these two worlds have the same algebraic structure, packaged as a category.\nFor an elliptic curve, this means that closed curves on a torus correspond to vector bundles and sheaves on the mirror elliptic curve.\n",
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      "first_stated": "1994",
      "posed_by": "Maxim Kontsevich"
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