Differentials of Cox rings: Jaczewski's theorem revisited

Abstract

A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V ⊗ \mathcal{O}_X by the sheaf of differentials Ω_X, given by the inclusion of a linear space V ⊂ Ext¹_X(\mathcal{O}_X, Ω_X). For Λ, a lattice of Cartier divisors, let ℛ_Λ denote the corresponding sheaf associated to V spanned by the first Chern classes of divisors in Λ. We prove that any projective, smooth variety on which the bundle ℛ_Λ splits into a direct sum of line bundles is toric. We describe the bundle ℛ_Λ in terms of the sheaf of differentials on the characteristic space of the Cox ring, provided it is finitely generated. Moreover, we relate the finiteness of the module of sections of ℛ_Λ and of the Cox ring of Λ.