We prove the Feldman--Ilmanen--Knopf conjecture for finite time Type I singularities of the K\"ahler--Ricci flow. More precisely, for any compact K\"ahler manifold $Y$ and its blow-up $\pi:\operatorname{Bl}_pY\longrightarrow Y$, if $[\omega_0]-Tc_1(M)=\pi^*[\omega_Y]$, then any Type I parabolic blow-up limit of the K\"ahler Ricci flow along the exceptional divisor is the FIK shrinker $\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1))$.
We study collapsing finite-time singularities of the unnormalized K\"ahler--Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle $X^n\rightarrow Y^m$, we prove a maximal splitting theorem by the K\"ahler-Ricci flow that every tangent space at a fixed limiting point splits globally...
Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion from a K\"ahler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\in\Delta$, then $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This gives a partial answer to Siu's conjecture on the invariance of plurigene...
F. Deng, Jia-Fu Ning, Zhi-Wei Wang et al.· 2 citations· ⚡1
The analytic minimal model program proposed in \cite{SongTianSingularities, JST} seeks to describe the birational transitions and fibre collapsing of the K\"ahler--Ricci flow through the geometry of its singularities. A fundamental conjecture of \cite{JST} predicts Type I bounds for the scalar curvature and diameter of...
We study the singularity type and models of the K\"ahler--Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of K\"ahler--Einstein manifolds $N := N_1 \times \cdots \times N_r$, with metric constructed using the ansatz considered in \cite{DW2011}, \cite{WW} et....
Let $3\leq n\leq7$, $2\leq k\leq n-1$, and $m=n-k-1$. We prove that a complete, connected, noncompact spin manifold $(M^n,g)$ with scalar curvature $R_M\geq k(k-1)$ is isometric to $\mathbb{S}^k\times\mathbb{T}^m_\Lambda\times\mathbb{R}$ if it admits a smooth proper map of nonzero degree to $\mathbb{S}^k\times\mathbb{T...
Let $(M^{2m},g,f,J)$ be a complete nonsteady gradient K\"ahler-Ricci soliton satisfying $\mathrm{Ric}+\nabla^2 f=\lambda g$, $\lambda\neq 0$. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as $N^{2k}\times\mathbb{C}^{m...
M. Horácio, Wen-Qi Li, Jian-Yu Ou et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
