Free affine actions of unipotent groups on \(\mathbb {C}^{n}\) (Q882642)

The author considers free affine algebraic actions \(\varphi: V\times X\rightarrow X\) of unipotent complex algebraic groups \(V\) on \(X=\mathbb C^n\) with regard to the existence of analytic or algebraic geometric quotients and their properties. In the above setting the properness of the action is necessary and sufficient for the existence of an analytic geometric quotient. There is a system \(\{x_1,\dots,x_n\}\) of global coordinates on \(X\) such that the \((n+1)\)-dimensional complex vector space \(W\subset\mathbb C[X]\) generated by \(\{1,x_1,\dots,x_n\}\) is \(V\)-invariant, and associated to \(\varphi\) is a canonical homomorphism \(\rho\) from the Lie algebra \({\mathbf v}\) of \(V\) into the Lie algebra of locally nilpotent algebraic derivations of \(\mathbb C[X]\). The degree deg \(\varphi\) of the action \(\varphi\) is by definition the smallest integer \(k\) such that for any \(k+1\) elements \(\delta_0,\dots,\delta_k\in\rho({ \mathbf v})\) the equation \(\delta_0\circ\dots\circ\delta_k=0\) holds on \(W\). Using the fact that a free affine action of the abelian group \((\mathbb C,+)\) on \(X\) is generated by a special locally nilpotent derivation of \(\mathbb C[X]\), it is shown that such an action admits a global slice and an algebraic geometric quotient which is isomorphic to \(\mathbb C^{n-1}\). The quotient of a free algebraic triangular action of \((\mathbb C,+)\) on \(X\) is in general not isomorphic to \(\mathbb C^{n-1}\), see the example of \textit{J. Winkelmann} [Math. Ann. 286, No. 1--3, 593--612 (1990; Zbl 0708.32004)]. A main result of the paper states that the action \(\varphi\) admits an analytic geometric quotient if deg \(\varphi\leq 2\) and even an algebraic geometric quotient isomorphic to \(\mathbb C^{n-\dim V}\) if deg \(\varphi\leq 1\). An example is given of a free affine \((\mathbb C^2,+)\)-action on \(\mathbb C^5\) of degree \(3\) where the quotient is not Hausdorff. The author then considers free affine algebraic actions \(\varphi\) of \((\mathbb C^2,+)\) on \(X\). They have algebraic geometric quotients isomorphic to \(\mathbb C^{n-2}\) if they admit a semi-invariant (an element \(w\in W\) with \(\mathbb C^2(w)=\{w+a\cdot 1\mid a\in\mathbb C\}\)). For \(n\geq 5\) there are only two actions (up to an affine change of coordinates) \(\varphi_j\), \(0\leq j\leq 1,\) with deg \(\varphi_j=n-2\) and no semi-invariants. One of them is generic, say \(\varphi_0\). The quotients of these actions have the following property: For \(n+j\geq 5\) odd, \(\varphi_j\) has an algebraic geometric quotient isomorphic to \(\mathbb C^{n-2}\), and for \(n+j\geq 6\) even, the quotient of \(X\) with respect to the action \(\varphi_j\) is not Hausdorff.