Characterizing maximal varieties via Bredon cohomology

We obtain a characterization of Maximal and Galois-Maximal C 2 -spaces (including real algebraic varieties) in terms of RO ( C 2 ) -graded cohomology with coefficients in the constant Mackey functor F _ 2 , using the structure theorem of Clover May. Other known characterizations, for instance in terms of equivariant Borel cohomology, are also rederived from this. For the particular case of a smooth projective real variety V , equivariant Poincaré duality is used to deduce further symmetry restrictions for the decomposition of the RO ( C 2 ) -graded cohomology of the complex locus V ( C ) given by the same structure theorem. We illustrate this result with some computations, including the RO ( C 2 ) -graded cohomology with F _ 2 coefficients of real K 3 surfaces.