Homological mirror symmetry of CPn and their products via Morse homotopy

Abstract

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many examples, the derived category Db(coh(X)) of coherent sheaves on a toric manifold X is compared with the Fukaya–Seidel category of the Milnor fiber of the corresponding Landau–Ginzburg potential. We instead consider the dual torus fibration π: M → B of the complement of the toric divisors in X, where B̄ is the dual polytope of the toric manifold X. A natural formulation of homological mirror symmetry in this setup is to define Fuk(M̄) a variant of the Fukaya category and show the equivalence Db(coh(X))≃Db(Fuk(M̄)). As an intermediate step, we construct the category Mo(P) of weighted Morse homotopy on P≔B̄ as a natural generalization of the weighted Fukaya–Oh category proposed by Kontsevich and Soibelman [Symplectic geometry and mirror symmetry (Seoul, 2000) (World Science Publishing, 2001), p. 203]. We then show that a full subcategory MoE(P) of Mo(P) generates Db(coh(X)) for the cases X is a complex projective space and their products.