Homological mirror symmetry for weighted projective spaces and Morse Homotopy

TL;DR

This paper extends homological mirror symmetry to weighted projective spaces and toric orbifolds, using Morse homotopy on the base space of dual torus fibrations, connecting symplectic and algebraic geometry.

Contribution

It generalizes the weighted Morse homotopy approach to toric orbifolds, establishing mirror symmetry for weighted projective spaces.

Findings

Established homological mirror symmetry for weighted projective spaces.

Extended Morse homotopy categories to toric orbifolds.

Connected derived categories of coherent sheaves with Morse homotopy categories.

Abstract

Kontsevich and Soibelman discussed homological mirror symmetry by using the SYZ torus fibrations, where they introduced the weighted version of Fukaya-Oh's Morse homotopy on the base space of the dual torus fibration in the intermediate step. Futaki and Kajiura applied Kontsevich-Soibelman's approach to the case when a complex manifold $X$ is a smooth compact toric manifold. There, they introduced the category of weighted Morse homotopy on the moment polytope of toric manifolds, and compared this category to the derived category of coherent sheaves on $X$ instead of the Fukaya category. In this paper, we extend their setting to the case of toric orbifolds, and discuss this version of homological mirror symmetry for weighted projective spaces.