On unipotent quotients and some \(\mathbb A^1\)-contractible smooth schemes.

*(English)*Zbl 1157.14032In topology it is known that every open contractible (smooth or PL) manifold of dimension \(n\geq 4\) can be realized as a quotient of some \(\mathbb{R}^{n+k}\) by some free \(\mathbb{R}^k\)-action. It is also known that there are many non-trivial such manifolds (“non-trivial”= not diffeomorphic to \(\mathbb{R}^n\)).

This raises the question for an algebro-geometric analogue. Asok and Doran investigate such an analogue in the framework of “motivic homotopy”-category in the sense of Morel and Voevodsky using “Nisnevich sheaves”. They ask for “\(\mathbb{A}^1\)-contractible” algebraic varieties. For a complex algebraic variety being “\(\mathbb{A}^1\)-contractible” implies that the underlying complex space is contractible in the usual sense. However, the notion “\(\mathbb{A}^1\)-contractible” makes sense for arbitrary ground fields.

A stability notion for action of unipotent groups is introduced and related to freeness of the action and the existence of quotients.

Based on this, it is shown that large families of non-isomorphic \(\mathbb{A}^1\)-contractible varieties can be obtained from actions of unipotent groups induced by linear representations. More precisely, for every \(m\geq 6\) there are families of arbitrarily large dimensions parametrizing pairwise non-isomorphic \(\mathbb{A}^1\)-contractible varieties of dimension \(m\). Furthermore, these \(\mathbb{A}^1\)-contractible varieties are quasi-affine but not affine.

This raises the question for an algebro-geometric analogue. Asok and Doran investigate such an analogue in the framework of “motivic homotopy”-category in the sense of Morel and Voevodsky using “Nisnevich sheaves”. They ask for “\(\mathbb{A}^1\)-contractible” algebraic varieties. For a complex algebraic variety being “\(\mathbb{A}^1\)-contractible” implies that the underlying complex space is contractible in the usual sense. However, the notion “\(\mathbb{A}^1\)-contractible” makes sense for arbitrary ground fields.

A stability notion for action of unipotent groups is introduced and related to freeness of the action and the existence of quotients.

Based on this, it is shown that large families of non-isomorphic \(\mathbb{A}^1\)-contractible varieties can be obtained from actions of unipotent groups induced by linear representations. More precisely, for every \(m\geq 6\) there are families of arbitrarily large dimensions parametrizing pairwise non-isomorphic \(\mathbb{A}^1\)-contractible varieties of dimension \(m\). Furthermore, these \(\mathbb{A}^1\)-contractible varieties are quasi-affine but not affine.

Reviewer: Jörg Winkelmann (Bayreuth)