Kähler-Ricci flow of cusp singularities on quasi projective varieties

Abstract

Let $\overline{M}$ be a compact complex manifold with smooth Kähler metric $\eta$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hat{\omega }$ be a Carlson-Griffiths type metric on $M$. We study complete solutions to Kähler-Ricci flow on $M$ which are comparable to $\hat{\omega }$, starting from a smooth initial metric ${\omega }_0=\eta +i\partial \bar{\partial }{\varphi }_0$ where ${\varphi }_0\in C^{\infty }(M)$. When ${\omega }_0\ge c\hat{\omega }$ on $M$ for some $c>0$ and ${\varphi }_0$ has zero Lelong number, we construct a smooth solution $\omega (t)$ to Kähler-Ricci flow on $M\times [0,T_{[{\omega }_0]})$ where $T_{[{\omega }_0]}:=\sup \{T:[\eta ]+T(c_1(K_{\overline{M}})+c_1({\mathcal{O}}_D))\in {\mathcal{K}}_M\}$ so that $\omega (t)\ge (\frac{1}{n}-\frac{4\hat{K}t}{c})\hat{\omega }$ for all $t\le \frac{c}{4n\hat{K}}$ where $\hat{K}$ is a non-negative upper bound on the bisectional curvatures of $\hat{\omega }$ (see Theorem 1.2). In particular, we do not assume ${\omega }_0$ has bounded curvature. If ${\omega }_0$ has bounded curvature and is asymptotic to $\hat{\omega }$ in an appropriate sense, we construct a complete bounded curvature solution on $M\times [0,T_{[{\omega }_0]})$ (see Theorem 1.3). These generalize some of the results of Lott-Zhang in [15]. On the other hand if we only assume ${\omega }_0\ge c\eta$ on $M$ for some $c>0$ and ${\varphi }_0$ is bounded on $M$, we construct a smooth solution to Kähler-Ricci on $M\times [0,T_{[{\omega }_0]})$ which is equivalent to $\hat{\omega }$ for all positive times. This includes as a special case when ${\omega }_0$ is smooth on $\overline{M}$ in which case the solution becomes instantaneously complete on $M$ under Kähler-Ricci flow (see Theorem 1.1).

29 pages; corrections made to earlier version (see (3.8), also see remarks 5, 6)