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

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