Nef Hattori classes and products of projective lines

Source abstract

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 $\pm1$. 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.