Bigraded invariants for real curves

Abstract

For a proper smooth real algebraic curve $\Sigma$ we compute the ring structure of both its ordinary bigraded $Gal\left(ℂ∕ℝ\right)$–equivariant cohomology [Bull. Amer. Math. Soc. 4 (1981) 208–212] and its integral Deligne cohomology for real varieties [Math. Ann. 350 (2011) 973–1022]. These rings reflect both the equivariant topology and the real algebraic structure of $\Sigma$ and they are recipients of natural transformations from motivic cohomology. We conjecture that they completely detect the motivic torsion classes.