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Affine \( \mathbb{T} \)-varieties of complexity one and locally nilpotent derivations

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Abstract

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \( \mathbb{T} \) of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \( {\mathbb{Z}^n} \)-graded domain A, so that ∂ generates a k +-action on X that is normalized by the \( \mathbb{T} \)-action.

We provide a complete classiffication of pairs (X, ∂) in two cases: for toric varieties (n = dim X) and in the case where n = dim X − 1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular, we exhibit a family of nonrational varieties with trivial Makar-Limanov invariant.

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  1. Institut Fourier, Université Joseph Fourier, BP 74 38402 St., Martin d’Hères, France

    Alvaro Liendo

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  1. Alvaro Liendo
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Liendo, A. Affine \( \mathbb{T} \)-varieties of complexity one and locally nilpotent derivations. Transformation Groups 15, 389–425 (2010). https://doi.org/10.1007/s00031-010-9089-2

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