Nef Hattori classes and products of projective lines

Let $X$ be a smooth connected complex projective variety with an effective algebraic ${\mathbb{C}}^{\times }$-action, a nonempty finite fixed locus whose weights are $\pm 1$. We show that if its Hattori line bundles are nef, then $X$ is algebraically isomorphic to $({\mathbb{P}}^1{)}^{\dim X}$, and these bundles become the hyperplane bundles of the factors. The product statement follows from the nef-complexity classification of Enwright and collaborators, and we describe the isomorphism through global sections. To test the required nefness we specialise the orbit-graph criterion of Romano and Wiśniewski. The Hirzebruch surface ${\mathbb{F}}_2$ is an example why the topological hypotheses alone do not imply either the product conclusion or the Fano property.