##
**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.

Reviewer: Hans U. Boden (Hamilton/Ontario)

### 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 |