Strominger-Yau-Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category.
HAYATO NAKANISHIA.2. Compositions of morphisms in Mo E (P ) for Bl 3 (CP 2 ). 43 References 501. Introduction.Mirror symmetry has fascinated many physicists and mathematicians. Mirror symmetry was first observed in Calabi-Yau manifolds. It has been extended to cases of non Calabi-Yau, e.g., Fano manifolds. Homological mirror symmetry conjectured by Kontsevich [13] is a categorical formulation of mirror symmetry. This conjecture is a statement about the equivalence of two categories: the Fukaya category F uk(X) of a Calabi-Yau manifold X and the derived category D b (Coh( X)) of the coherent sheaves on a mirror Calabi-Yau manifold X. More precisely, the triangulated category induced by the Fukaya category is equivalent to D b (Coh( X)). In general, we can construct a triangulated category from a given A ∞ category [2, 13]. When two triangulated categories are induced by two A ∞ -quasi-isomorphic A ∞ categories, then the triangulated categories are equivalent to each other [17]. Therefore, when D b (Coh( X)) is obtained from an A ∞ category, we can discuss the homological mirror symmetry conjecture by considering whether underlying A ∞ categories are A ∞ -quasi-isomorphic to each other.Strominger-Yau-Zaslow proposed a construction of mirror pair in [18]. When a given Calabi-Yau manifold is equipped with a torus fibration, we can obtain a mirror Calabi-Yau manifold as a dual torus fibration. Kontsevich-Soibelman [14] proposed a framework to systematically prove the homological mirror symmetry via Morse homotopy for dual torus fibrations over a closed manifold without singular fibers. Fukaya-Oh [6] proved that the category Mo(B) of Morse homotopy on B is equivalent to the Fukaya category F uk(T * B) of the cotangent bundle T * B, and Kontsevich-Soibelman's approach is based on this result.Fano surfaces are also called del Pezzo surfaces, and the homological mirror symmetry of toric del Pezzo surfaces is discussed by Auroux-Katszarkov-Orlov [1] and Ueda [19]. In these papers, they consider a Fukaya-Seidel category [17] corresponding to a Landau-Ginzburg potential of toric del Pezzo surfaces as the symplectic side.In this paper, we apply SYZ construction to toric Fano surfaces as complex manifolds and discuss the homological mirror symmetry. In this situation, we consider Morse homotopy of the moment polytope instead of the Fukaya category. SYZ picture is useful to express the mirror functor explicitly, which is why we use SYZ construction instead of the Landau-Ginzburg potential. 9] proved the homological mirror symmetry for CP 2 , CP 1 × CP 1 and F 1 . They defined the category Mo(P ) of weighted Morse homotopy as a generalization of the weighted Fukaya-Oh category given in [14] to the case that the base manifo...
