{
  "abstract": "We study the singularity type and models of the Kähler--Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler--Einstein manifolds $N := N_1 \\times \\cdots \\times N_r$, with metric constructed using the ansatz considered in \\cite{DW2011}, \\cite{WW} et. al. In the earlier work \\cite{FT} by the authors, we considered the ``two-bolt'' case where both ends of the foliation close with the ``bolt'' $N$. In this article, we continue our work on the more subtle ``nut-bolt'' and ``two-nut'' cases. The former has one end of the interval closes with a nut-type collapse (i.e. $N' := N_2 \\times \\cdots \\times N_r$) and the other with a bolt (i.e. $N$). The compactification $\\widehat{M}$ is then a $\\mathbb{CP}^{m+1}$-bundle over $N'$. The ``two-nut'' case is one that both ends close with nut-type collapses, necessarily two of the $N_i$'s must be $\\mathbb{CP}^{m_0}$ and $\\mathbb{CP}^{m_\\ell}$, and the compactification $\\widehat{M}$ is a $\\mathbb{CP}^{m_0+m_\\ell+1}$-bundle over $\\prod_{k\\geq 3}N_k$. We proved that in all ``two-bolt'', ``nut-bolt'' and ``two-nut'' caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be $(Σ^{m+1}, g_Σ(t)) \\times (\\mathbb{C}^{k}, \\textrm{flat})$ with $m, k \\geq 0$, where $Σ$ is one of the following: $\\mathbb{CP}^{m+1}$, $\\textrm{Tot}(\\mathcal{L}^{\\oplus(m+1)})$, or a projectivization $\\mathbb{P}\\big(\\mathcal{O}^{\\oplus(m_0+1)} \\oplus \\mathcal{L}^{\\oplus(m_\\ell+1)}\\big)$ with $m_0 + m_\\ell = m$, and $\\mathcal{L}$ is a line bundle over the product of \\emph{some} of the $N_1, \\cdots, N_r$ factors. The metric $g_Σ(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03332",
  "author_total": 2,
  "authors": [
    {
      "name": "Frederick Tsz-Ho Fong"
    },
    {
      "name": "Hung Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/efqw8j0q1d1y4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03332",
  "fulltext_url": null,
  "id": "efqw8j0q1d1y4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03332",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7208715006",
  "pdf_url": "https://arxiv.org/pdf/2609.03332",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/efqw8j0q1d1y4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03332",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03332"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Singularity Models of Finite-Time Kähler-Ricci Flows",
  "title_text": "Singularity Models of Finite-Time Kähler-Ricci Flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/efqw8j0q1d1y4",
  "venue": "arXiv",
  "versions": []
}