On the Motivic Fundamental Groups of Smooth Toric Varieties

In this short note, we want to investigate combinatorial descriptions of the motivic fundamental group πA 1 1 (X(∆)) for a smooth toric variety associated to the fan ∆. This reduces the computation of the fundamental group of the toric variety X(∆) to a computation of the A1-localization of an explicitly given sheaf of groups. As a corollary, a smooth toric variety for which the irrelevant ideal in the Cox ring has codimension ≥ 3 has a torus as A1-fundamental group.