{
  "id": "2310.17554",
  "version": "v1",
  "published": "2023-10-26T16:46:02.000Z",
  "updated": "2023-10-26T16:46:02.000Z",
  "title": "Characterizing maximal varieties via Bredon cohomology",
  "authors": [
    "Pedro F. dos Santos",
    "Carlos Florentino",
    "Javier Orts"
  ],
  "comment": "21 pages, 5 figures",
  "categories": [
    "math.AG",
    "math.AT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-26T16:46:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14P25",
      "14F45",
      "55N91"
    ],
    "keywords": [
      "characterizing maximal varieties",
      "bredon cohomology",
      "graded cohomology",
      "structure theorem",
      "real algebraic varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}