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Non-linearizable algebraic \(k^*\)-actions on affine spaces. (English) Zbl 0933.14027
Let \(k^{[n]}\) be a polynomial ring in \(n\) variables over a field \(k\). The linearity problem of torus actions asks if any faithful algebraic \((k^*)^r\)-action of a torus on a \(k\)-affine space \(\mathbb{A}^n (=\text{Spec} k^{[n] })\) is always linearizable, i.e. linear after a suitable change of the coordinate system of \(\mathbb{A}^n\). The main purpose of the present paper is to give non-linearizable algebraic \((k^*)^r\)-actions of torus on \(\mathbb{A}^n\) in the following two cases:
(a) \(k\) is an infinite field of characteristic \(p>0\) for \(n>3\) and \(r=1,\dots,n-2\),
(b) \(k\) is the real field \(\mathbb{R}\) for \(n>4\) and \(r=1,\dots,n-3\).
As an application of the proof of these results we also give examples of nonsingular affine surfaces \(V\) and \(W\) such that \(V\times \mathbb{A}^1\cong_kW\times \mathbb{A}^1\), but \(V\ncong_kW\) for an arbitrary field \(k\). Moreover if \(k\) is algebraically closed, we show that some pairs of such \(V\) and \(W\) satisfy not only \(X\ncong_k Y\) but also \(\operatorname{Aut}_k X\ncong \operatorname{Aut}_kY\), where \(\operatorname{Aut}_kX\) and \(\operatorname{Aut}_kY\) denote the groups of all \(k\)-automorphisms of \(X\) and \(Y\), respectively.
Reviewer: T.Asanuma (Toyama)

14L30 Group actions on varieties or schemes (quotients)
20G05 Representation theory for linear algebraic groups
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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