Automorphism groups of Koras-Russell threefolds of the first kind

TL;DR

This paper investigates the automorphism groups of Koras-Russell threefolds of the first kind, extending previous results on the Russell cubic to a broader class of similar hypersurfaces.

Contribution

It generalizes the understanding of automorphism groups from the Russell cubic to all Koras-Russell threefolds of the first kind.

Findings

Automorphisms extend to ambient space

Existence of inequivalent isomorphic hypersurfaces

Generalization of previous automorphism results

Abstract

Koras-Russell threefolds are certain smooth contractible complex hypersurfaces in affine complex four-space which are not algebraically isomorphic to affine three-space. One of the important examples is the cubic Russell threefold, defined by the equation x^2y+z^2+t^3+x=0. In a previous article by A. Dubouloz, P.M. Poloni and the author, the automorphism group of the Russell cubic threefold was studied. It was shown, in particular, that all automorphisms of this hypersurface extend to automorphisms of the ambient space. This has several interesting consequences, including the fact that one can find another hypersurface which is isomorphic to the Russell cubic, but such that the two hypersurfaces are inequivalent. In the present article, we will discuss how some of these results can be generalized to the class of Koras-Russell threefolds of the first kind.