Nef Hattori classes and products of projective lines

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 Wi\'sniewski. 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.