Torsion of differentials on toric varieties
For a commutative semigroup \(S\) with \(0\) satisfying the cancellation property \(a+s=b+ s\Rightarrow a=b\) (for all \(a,b,s\in S)\), the author defines certain abelian groups \(T_l\) for each \(l\in S\) in such a way that the following holds: Suppose \(S\) is the set of lattice points of a rational polyhedral cone, and denote by \(\mathbb{C} [S]\) the semigroup algebra of \(S\) over the field \({\mathbb{C}}\) of complex numbers. Then \((\bigoplus_{l\in S} T_l)\otimes_\mathbb{Z} \mathbb{C}\) coincides with the torsion \(\mathbb{C}[S]\)-module of the sheaf \(\Omega^1_Y\) of Kähler differentials of the affine toric variety \(Y:= \text{Spec}(\mathbb{C} [S])\). The author computes the \(T_l\)'s when \(Y\) is a 2-dimensional affine toric variety, that is, a 2-dimensional cyclic quotient singularity.
scientific article; zbMATH DE number 895853