Homological mirror symmetry of $\mathbb{C}P^n$ and their products via Morse homotopy

TL;DR

This paper develops a new approach to homological mirror symmetry for toric manifolds by using Morse homotopy categories on dual torus fibrations, establishing equivalences with derived categories of coherent sheaves.

Contribution

It introduces a Morse homotopy category framework for understanding homological mirror symmetry in toric manifolds, extending previous Landau-Ginzburg potential methods.

Findings

Constructed the category Mo(P) of weighted Morse homotopy.

Showed that a subcategory of Mo(P) generates the derived category of coherent sheaves.

Applied the framework to complex projective spaces and their products.

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.