Kähler-Ricci flow of cusp singularities on quasi projective varieties

Abstract

Let $\overline{M}$ be a compact complex manifold with smooth Kähler metric $η$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hatω$ be a Carlson-Griffiths type metric on $M$. We study complete solutions to Kähler-Ricci flow on $M$ which are comparable to $\hatω$, starting from a smooth initial metric $ω_0=η+i\partial \bar{\partial} ϕ_0$ where $ϕ_0\in C^{\infty}(M)$. When $ω_0\geq c \hatω$ on $M$ for some $c>0$ and $ϕ_0$ has zero Lelong number, we construct a smooth solution $ω(t)$ to Kähler-Ricci flow on $M\times [0, T_{[ω_0 ]})$ where $T_{[ω_0 ]}:= \sup \{ T: [η] +T (c_1(K_{\overline{M}}) + c_1(\mathcal{O}_D))\in \mathcal{K}_M \}$ so that $ω(t)\geq (\frac{1}{n} - \frac{4\hat{K}t}{c} )\hatω$ for all $t\leq \frac{c}{4n\hat{K}}$ where $\hat{K}$ is a non-negative upper bound on the bisectional curvatures of $\hatω$ (see Theorem 1.2). In particular, we do not assume $ω_0$ has bounded curvature. If $ω_0$ has bounded curvature and is asymptotic to $\hatω$ in an appropriate sense, we construct a complete bounded curvature solution on $M\times [0, T_{[ω_0 ]})$ (see Theorem 1.3). These generalize some of the results of Lott-Zhang in [15]. On the other hand if we only assume $ω_0\geq c η$ on $M$ for some $c>0$ and $ϕ_0$ is bounded on $M$, we construct a smooth solution to Kähler-Ricci on $M\times [0, T_{[ω_0 ]})$ which is equivalent to $\hatω$ for all positive times. This includes as a special case when $ω_0$ is smooth on $\overline{M}$ in which case the solution becomes instantaneously complete on $M$ under Kähler-Ricci flow (see Theorem 1.1).