Paper detail

Differentials of Cox rings: Jaczewski's theorem revisited

A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V \otimes O_X by the sheaf of differentials Ω_X, given by the inclusion of a linear space V in Ext^1(O_X,Ω_X). For Λ, a lattice of Cartier divisors, let R_Λdenote 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_Λsplits into a direct sum of line bundles is toric. We describe the bundle R_Λin 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_Λand of the Cox ring of Λ.