{
  "abstract": "Let $(M,J)$ be a compact almost hermitian $4$-manifold with a smooth embedded $J$-holomorphic curve $C$ representing the canonical class. Motivated by the symplectic Bogomolov--Miyaoka--Yau conjecture, we choose a twisted spin$^{\\text{c}}$ Dirac operator ${\\mathcal D}$ on $M$ satisfying $$ \\operatorname{ind}{\\mathcal D}=3c_2(M)-c_1^2(M). $$ We then use the curve $C$ to construct a complex-linear perturbation ${\\mathcal A}$ of ${\\mathcal D}$ whose singular set is $Z_α\\sqcup C$, where $Z_α=α^{-1}(0)$ for a transverse section $α$ of $Λ^{1,0}M$ nonvanishing on $C$. Applying Maridakis' index localization theorem, we express $\\operatorname{ind}{\\mathcal D}$ as the sum of localized contributions: $3c_2(M)$ from $Z_α$ and $-c_1^2(M)$ from $C$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07177",
  "author_total": 1,
  "authors": [
    {
      "name": "Junho Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26sda881cfgxz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07177",
  "fulltext_url": null,
  "id": "26sda881cfgxz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07177",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7212053486",
  "pdf_url": "https://arxiv.org/pdf/2609.07177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26sda881cfgxz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.07177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Localizing a Dirac operator via $J$-holomorphic curves",
  "title_text": "Localizing a Dirac operator via J-holomorphic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/26sda881cfgxz",
  "venue": "arXiv",
  "versions": []
}