Real forms of some Gizatullin surfaces and Koras-Russell threefolds

TL;DR

This paper classifies the real forms of certain algebraic surfaces and threefolds, revealing the number of distinct real forms based on polynomial degree and symmetry, with some varieties having multiple forms and others only one.

Contribution

It provides a detailed classification of real forms for Gizatullin surfaces and Koras-Russell threefolds, highlighting conditions for the number of forms.

Findings

Gizatullin surfaces have 0, 2, 3, 4, or 6 real forms depending on polynomial properties.

Koras-Russell threefolds have exactly one real form up to isomorphism.

The classification depends on polynomial degree and symmetry considerations.

Abstract

We describe the real forms of Gizatullin surfaces of the form $xy=p(z)$ and of Koras-Russell threefolds of the first kind. The former admit zero, two, three, four or six isomorphism classes of real forms, depending on the degree and the symmetries of the polynomial~$p$. The latter, which are threefolds given by an equation of the form $x^dy+z^k+x+t^\ell=0$, all admit exactly one real form up to isomorphism.