  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.
