{
  "abstract": "We study the Kontsevich--Zorich cocycle, its fixed tensor constructions, and their finite direct sums over products of strata of Abelian differentials. The acting groups are closed subgroups $G\\subset(\\mathrm{SL}_2\\mathbb R)^n$ that, up to factor permutations and coordinatewise conjugation by rotations, are products of diagonal $\\mathrm{SL}_2\\mathbb R$ blocks and a finite central group. For ergodic invariant probability measures, we prove Deligne-type semisimplicity: the bundles admit Hodge-compatible decompositions into $G$-invariant pieces $V_i\\otimes_{A_i}W_i$, where $A_i$ are division algebras, $V_i$ carry pure variations of Hodge structure, and the multiplicity spaces $W_i$ carry finite direct sums of pure Hodge structures. Every invariant subbundle arises from $A_i$-submodules of the $W_i$. If block marginal measures have smooth local supports and are locally equivalent to smooth volume, Hodge-orthogonal projections onto invariant subbundles are locally polynomial in period coordinates on the normalized support. The proof obtains a local product-of-quadrics model and joint polynomiality from polynomiality along stable and unstable leaves. On the cone obtained by scaling each diagonal block together, projection coefficients admit homogeneous degree-zero expressions with integral powers of reciprocal block areas. For the full product group $(\\mathrm{SL}_2\\mathbb R)^n$, single-stratum measure classification supplies the support hypothesis, yielding rigidity for every ergodic invariant probability measure. Under the transported support hypothesis, arbitrary coordinatewise $\\mathrm{SL}_2\\mathbb R$-conjugates also admit polynomial equivariant projectors, though not necessarily Hodge-orthogonal. Genus-two examples show that nonrotational conjugation can destroy Deligne-torus invariance, and independent factor scalings can require square roots of area functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21247",
  "author_total": 1,
  "authors": [
    {
      "name": "Polina Baron"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vedtk73763yq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21247",
  "fulltext_url": null,
  "id": "3vedtk73763yq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21247",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7222669909",
  "pdf_url": "https://arxiv.org/pdf/2610.21247",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vedtk73763yq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21247",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21247"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Semisimplicity and Rigidity of the Kontsevich--Zorich Cocycle for Products of Strata",
  "title_text": "Semisimplicity and Rigidity of the Kontsevich--Zorich Cocycle for Products of Strata",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/3vedtk73763yq",
  "venue": "arXiv",
  "versions": []
}