SYZ mirror of Hirzebruch surface ${\mathbb{F}}_k$ and Morse homotopy

We study homological mirror symmetry for Hirzebruch surface ${\mathbb{F}}_k$ as a complex manifold by using the Strominger-Yau-Zaslow construction of mirror pair and Morse homotopy. For the toric Fano surfaces, Futaki-Kajiura and the author proved homological mirror symmetry by using Morse homotopy in arXiv:2008.13462, arXiv:2012.06801, and arXiv:2303.07851. In this paper, we extend Futaki-Kajiura’s result of Hirzebruch surface ${\mathbb{F}}_1$ to ${\mathbb{F}}_k$. We discuss Morse homotopy and show homological mirror symmetry in the sense above holds true.