The polar group of a real form of an affine or projective $\mathbb{C}$-variety

TL;DR

This paper introduces the polar group as an invariant to classify real forms of complex varieties, providing calculations for specific curves like the real line and algebraic 1-sphere.

Contribution

It defines the polar group for real forms of complex varieties and demonstrates its effectiveness in distinguishing different real forms.

Findings

Polar groups are computed for several curves.

The polar group encodes residual divisor information.

The invariant helps classify real forms up to isomorphism.

Abstract

A general problem is to classify the real forms of a complex variety up to isomorphism. This paper introduces the polar group of a real form $X$ of a complex variety $Y$ as a tool to distinguish such real forms. This group is an invariant of $X$ which encodes information about the residual divisors in the coordinate ring of $Y$ over the coordinate ring of $X$. We calculate polar groups for various curves, including the real line and the algebraic 1-sphere.