Uniqueness of asymptotically conical K\"ahler-Ricci flow

报告人：陈龙腾（巴黎萨克雷大学）

时 间：2025年8月10日（星期日），上午10:00-11:00

地 点：海纳苑2幢102

摘 要：A K\ahler cone appears as a normal algebraic variety with one isolated singular point, and the K\ahler-Ricci flow is expected to desingularize this singularity instantaneously. A precise example is given by Feldman, Ilmanen and Knopf in 2003. For any integers $k>n\ge 2$ and real number $p>0$, they constructed a forward self-similar solution to K\ahler-Ricci flow $g(t)_{t>0}$ on $\mathcal{O}(-k)$ (holomorphic line bundle over $\C\P^{n-1}$) such that outside the zero section, when $t$ tends to 0, such flow converges locally smoothly to the K\ahler cone $(\C^n/\Z^k, i\partial\bar\partial\left(\frac{|\cdot|^{2p}}{p}\right)$.

In 2019, Conlon, Deruelle and Sun generalize this result for any K\ahler cone that admits a smooth canonical model. Given a K\ahler cone $(C_0,g_0)$ with its smooth canonical model $M$, one can find a unique forward self-similar solution to K\ahler-Ricci flow $g(t)_{t>0}$ such that when $t$ tends to 0, $\pi_*g(t)$ converges to $g_0$ locally smoothly outside the apex, where $\pi: M\mapsto C_0$ is a K\ahler resolution.

In this talk, we will show that this desingularisation has a uniqueness property. Given $\tilde{g}(t)_{t\in (0,T)}$ a generic solution to K\ahler-Ricci flow which satisfies some conditions such that $\pi_*\tilde{g}(t)$ converges to $g_0$ locally smoothly when $t$ tends to $0$ outside the apex, then $\tilde{g}(t)=g(t)$ for all $t\in (0,T)$. Especially, among the conditions that we suppose, we only need a $\frac{C}{t}$ bound for the Ricci curvature tensor of $\tilde{g}$.

联系人：江文帅（wsjiang@zju.edu.cn)