Singularity Models of Finite-Time Kähler-Ricci Flows

@misc{fong2026singularity,
  title = {{Singularity Models of Finite-Time Kähler-Ricci Flows}},
  author = {Frederick Tsz-Ho Fong and Hung Tran},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03332},
  eprint = {2609.03332},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03332},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Singularity Models of Finite-Time Kähler-Ricci Flows
AU  - Frederick Tsz-Ho Fong
AU  - Hung Tran
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03332
UR  - https://arxiv.org/abs/2609.03332
AB  - 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 × ⋯ × 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 × ⋯ × N_r) and the other with a bolt (i.e. N). The compactification M̂ is then a 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 ℂℙ^m_0 and ℂℙ^m_ℓ, and the compactification M̂ is a ℂℙ^m_0+m_ℓ+1-bundle over ∏_k≥ 3N_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)) × (ℂ^k, flat) with m, k ≥ 0, where Σ is one of the following: ℂℙ^m+1, Tot(ℒ^⊕(m+1)), or a projectivization ℙ(𝒪^⊕(m_0+1) ⊕ ℒ^⊕(m_ℓ+1)) with m_0 + m_ℓ = m, and ℒ is a line bundle over the product of \emph{some} of the N_1, ⋯, N_r factors. The metric g_Σ(t) is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.
ER  -