Note on the candidate counter-example in the cancellation problem for affine spaces posed by Arno Van den Essen

TL;DR

This paper proves a specific case of the cancellation problem for affine spaces over complex domains, providing a negative answer to a candidate counter-example related to the conjecture by Van den Essen, and confirms several related problems remain open.

Contribution

It establishes a new criterion for when a complex affine domain is isomorphic to affine space, and applies this to disprove a candidate counter-example to Van den Essen's conjecture.

Findings

Proved a key property for complex affine domains related to the cancellation problem.

Disproved the candidate counter-example to Van den Essen's conjecture.

Confirmed that several major conjectures in affine algebraic geometry remain unresolved.

Abstract

We have proved the following Problem:{\it Let $R$ be a $\mathbb{C}$-affine domain, let $T$ be an element in $R \setminus \mathbb{C}$ and let $i : \mathbb{C}[T] \hookrightarrow R$ be the inclusion. Assume that $R/TR \cong_{\mathbb{C}} \mathbb{C}^{[n-1]}$ and that $R_T \cong_{\mathbb{C}[T]} \mathbb{C}[T]_T^{[n-1]}$. Then $R \cong_{\mathbb{C}} \mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {\it Let $A:=\mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be the polynomials in $\mathbb{C}[u]$. Let $D:= f'(u)\partial_x + g'(u)\partial_y + h'(u)\partial_z + t\partial_u$\ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D \not\cong_{\mathbb{C}} \mathbb{C}^{[4]}$.} Consequently our result in this short paper guarantees that the conjectures : "the Cancellation Problem for affine spaces", "the Linearization Problem", "the Embedding Problem" and "the affine $\mathbb{A}^n$-Fibration Problem" are still open.