Homological mirror symmetry of $\mathbb{C}P^n$ 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 example, the derived category $D^b(coh(X))$ 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 $\pi:M \to B$ of the complement of the toric divisors in $X$, where $\bar{B}$ is the dual polytope of the toric manifold $X$. A natural formulation of homological mirror symmetry in this set-up is to define $Fuk(\bar{M})$ a variant of the Fukaya category and show the equivalence $D^b(coh(X)) \simeq D^b(Fuk(\bar{M}))$. As an intermediate step, we construct the category $Mo(P)$ of weighted Morse homotopy on $P:=\bar{B}$ as a natural generalization of the weighted Fukaya-Oh category proposed by Kontsevich-Soibelman. We then show a full subcategory $Mo_{\mathcal{E}}(P)$ of $Mo(P)$ generates $D^b(coh(X))$ for the cases $X$ is a complex projective space and their products.