Hirzebruch surfaces in a one-parameter family

TL;DR

This paper introduces a one-parameter family of spaces encompassing all Hirzebruch surfaces, viewed through complex foliations and symplectic cuts, providing new geometric perspectives.

Contribution

It presents a unified family of spaces including all Hirzebruch surfaces, analyzed via complex foliations and symplectic cuts, offering novel geometric insights.

Findings

Unified family of spaces including all Hirzebruch surfaces

Two distinct geometric perspectives introduced

Potential applications in complex and symplectic geometry

Abstract

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 [BZ1]. Second, as a symplectic cut of the manifold $\mathbb{C}\times S^2$ in a possibly nonrational direction, following [BP2].