On uniqueness of additive actions on complete toric varieties (Q2168834)

An additive action on an algebraic variety \(X\) of dimension \(n\) over a field \(K\) is an effective regular action of the commutative unipotent group \(\mathbb{G}_a^n=\mathbb{G}_a\times\cdots\times \mathbb{G}_a\) on \(X\), with an open orbit. In the paper under review, the author gives a criterion for the uniqueness of additive actions on complete toric varieties. More precisely, for \(X\) a complete toric variety admitting an additive action, the main result shows that any two additive actions on \(X\) are conjugate by an automorphism of \(X\) if and only if a maximal unipotent subgroup of the automorphism group \(\mathrm{Aut}(X)\) is of the same dimension as \(X\). This criterion is obtained from explicit combinatorial conditions equivalent to the uniqueness of additive action in terms of Demazure roots of the toric variety \(X\).