{
  "abstract": "Let $m\\geq 2$ and $d\\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\\mathrm{MF}^{\\mathrm{gr}}(f)$ of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, $\\dim_{\\mathbb{Q}}\\operatorname{Hdg}\\bigl(\\mathrm{MF}^{\\mathrm{gr}}(f),\\mathbb{Q}\\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\\bigl((d-1)^{2m+2}+d-1\\bigr)/d+d-1$. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03784",
  "author_total": 2,
  "authors": [
    {
      "name": "Xun Lin"
    },
    {
      "name": "Shizhuo Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fj64ec78gx2qm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03784",
  "fulltext_url": null,
  "id": "fj64ec78gx2qm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03784",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7208773573",
  "pdf_url": "https://arxiv.org/pdf/2609.03784",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fj64ec78gx2qm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03784",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03784"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A note on the noncommutative Hodge conjecture for graded matrix factorizations",
  "title_text": "A note on the noncommutative Hodge conjecture for graded matrix factorizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/fj64ec78gx2qm",
  "venue": "arXiv",
  "versions": []
}