Automorphism groups of semi-rigid affine varieties

Prove that if X is a semi-rigid affine variety, then the identity component of its automorphism group satisfies Aut(X)° = T ⋉ U(X), where T is a maximal torus in Aut(X) and U(X) is the subgroup generated by all additive-group subgroups of Aut(X).

Background

A semi-rigid affine variety is one for which all additive-group actions have the same ring of invariants; equivalently, all such actions commute and are rational replicas of one another. The subgroup U(X) is generated by all additive-group subgroups in the automorphism group.

The conjecture generalizes the rigid-variety conjecture by predicting that the identity component of the automorphism group is generated by a maximal torus together with the subgroup generated by all additive-group actions. The paper states that the conjecture is known for toric varieties and proves it under additional hypotheses: normality, rationality, constant invertible global functions, finitely generated divisor class group, and the existence of a maximal torus of dimension dim X − 1.

References

If $X$ is a semi-rigid affine variety, then $\Aut(X)\circ=T\ltimes (X)$ where $T$ is a maximal torus in $\Aut(X)$.

— Automorphism groups of semi-rigid $T$-varieties of complexity one  (2609.09362 - Rassolov, 8 Sep 2026) in Introduction, Conjecture 2 (labelled Conjecture \ref{Semi-rigidConj})