Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms

TL;DR

This paper introduces a framework for understanding families of real forms of algebraic varieties and constructs examples of quasi-projective real surfaces with moduli spaces of real forms of arbitrarily high dimension.

Contribution

It provides a rigorous definition of families of real forms and constructs the first examples of quasi-projective real surfaces with large moduli spaces of real forms.

Findings

Constructed the first quasi-projective real surface with a moduli space of real forms of dimension at least n.

Extended constructions to higher-dimensional varieties with arbitrarily large moduli of real forms.

Established fundamental properties of the notion of families of real forms of varieties.

Abstract

We propose a framework to give a precise meaning to the intuitive notion of "family of real forms of a variety parametrised by a variety" and study some fundamental properties of this notion. As an illustration, for any $n \geq 1$, we construct the first example of a quasi-projective real surface whose mutually non-isomorphic real forms admit a moduli of dimension at least $n$, parametrised by the real points of an affine $n$-space. Expanding on these constructions, we can give quasi-projective real varieties of any dimension whose algebraic moduli of the non-isomorphic real forms has arbitrarily positive dimension.