Singularity Models of Finite-Time Kähler-Ricci Flows

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, et. al. In the earlier work 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. $N^{\prime }:=N_2\times \cdots \times N_r$) and the other with a bolt (i.e. $N$). The compactification $\hat{M}$ is then a ${\mathbb{CP}}^{m+1}$-bundle over $N^{\prime }$. 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 $\hat{M}$ is a ${\mathbb{CP}}^{m_0+m_{\ell }+1}$-bundle over ${\prod }_{k\ge 3}{N}_k$. 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 $({\Sigma }^{m+1},g_{\Sigma }(t))\times ({\mathbb{C}}^k,\text{flat})$ with $m,k\ge 0$, where $\Sigma$ is one of the following: ${\mathbb{CP}}^{m+1}$, $\text{Tot}({\mathcal{L}}^{\oplus (m+1)})$, or a projectivization $\mathbb{P}({\mathcal{O}}^{\oplus (m_0+1)}\oplus {\mathcal{L}}^{\oplus (m_{\ell }+1)})$ with $m_0+m_{\ell }=m$, and $\mathcal{L}$ is a line bundle over the product of some of the $N_1,\cdots ,N_r$ factors. The metric $g_{\Sigma }(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.