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Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. (English) Zbl 1204.14019

The homological mirror symmetry conjecture was first formulated by Kontsevich and gives an equivalence between the derived category of coherent sheaves on a Calabi-Yau manifold \(M\) and the derived Fukaya category of its mirror \(\check{M}\). The purpose of the present article, which is a continuation of [M. Abouzaid, Geom. Topol. 10, 1097–1156 (2006; Zbl 1160.14030)], is to provide evidence for Kontsevich’s extended homological mirror symmetry conjecture by studying the case of a smooth projective toric variety \(X\). The mirror of \(X\) is conjecturally a Laurent polynomial \(W: (\mathbb{C}^*)^n \to \mathbb{C}\) called the Landau-Ginzburg model. Deforming \(W\) to a family \(W_{t,s}\) of Lefschetz fibrations such that the pairs \(((\mathbb{C}^*)^n, W_{t,s}^{-1}(0))\) are symplectomorphic for different values of \(s\) and \(t\) and letting \(M\) denote any hypersurface of the form \(W_{t,s}^{-1}(0)\), the author constructs a Fukaya pre-category Fuk\(((\mathbb{C}^*)^n, M)\) consisting of compact Lagrangians on \((\mathbb{C}^*)^n\) whose boundary lies on \(M\). The main result in this paper (Theorem 1.2) is that Fuk\(((\mathbb{C}^*)^n, M)\) is quasi-equivalent as an \(A_\infty\) pre-category to the category of line bundles on \(X\). The proof is somewhat technical and will not be outlined here, but much of it involves establishing results at the \(A_\infty\) level which are more-or-less well understood at the level of homology.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
53D40 Symplectic aspects of Floer homology and cohomology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1160.14030
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