Hirzebruch surfaces in a one–parameter family

We introduce a family of spaces, parametrized by positive real numbers, that includes all of the Hirzebruch surfaces. Each space is viewed from two distinct perspectives. First, as a leaf space of a compact, complex, foliated manifold, following Battaglia and Zaffran (Int Math Res Not 22:11785–11815, 2015). Second, as a symplectic cut of the manifold C×S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb C\times S^2$$\end{document} in a possibly nonrational direction, following Battaglia and Prato (Int J Math 29:1850063, 2018).