Affine T-varieties of complexity one and locally nilpotent derivations

Let X = SpecA be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine Zn-graded domain A, so that ∂ generates a k+-action on X. We provide a complete classification of pairs (X,∂) in two cases: for toric varieties (n = dimX) and in the case where n = dimX − 1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we show that ker ∂ is finitely generated. Thus the generalized Hilbert’s fourteenth problem has a positive answer in this particular case, which strengthen a result of Kuroda. As another application, we compute the homogeneous Makar-Limanov invariant of such varieties. In particular we exhibit a family of non-rational varieties with trivial Makar-Limanov invariant.