Characterizing maximal varieties via Bredon cohomology

Author:

Pedro F. dos Santos, Carlos Florentino, Javier Orts

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Algebraic Topology (math.AT)

journal:

date:

2023-10-25 16:00:00

Abstract

We obtain a characterization of Maximal and Galois-Maximal $C_2$-spaces (including real algebraic varieties) in terms of $\operatorname{RO}(C_2)$-graded cohomology with coefficients in the constant Mackey functor $\underline{\mathbf{F}}_2$, 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 Poincar\'{e} duality from \cite{pedro&paulo:quaternionic_algebraic_cycles} is used to deduce further symmetry restrictions for the decomposition of the $\operatorname{RO}(C_2)$-graded cohomology of the complex locus $V(\mathbf{C})$ given by the same structure theorem. We illustrate this result with some computations, including the $\operatorname{RO}(C_2)$-graded cohomology with $\underline{\mathbf{F}}_2$ coefficients of real $K3$ surfaces.

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