Characterizing maximal varieties via Bredon cohomology

Pedro F. dos Santos; Carlos Florentino; Javier Orts

paper

Characterizing maximal varieties via Bredon cohomology

arXiv:2310.17554

Abstract

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 \cite{clover_may:structure_theorem}. 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 from \cite{pedro&paulo:quaternionic_algebraic_cycles} 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.

21 pages, 5 figures