K\"ahler-Ricci flow of cusp singularities on quasi projective varieties

Abstract

Let $\overline{M}$ be a compact complex manifold with smooth K\"ahler 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 K\"ahler-Ricci flow on $M$ which are comparable to $\hat{\omega}$, starting from a smooth initial metric $\omega_0=\eta +i\partial \bar{\partial} \phi_0$ where $\phi_0\in C^{\infty}(M)$. When $\omega_0\geq c \hat{\omega}$ on $M$ for some $c>0$ and $\phi_0$ has zero Lelong number, we construct a smooth solution $\omega(t)$ to K\"ahler-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 $\omega(t)\geq (\frac{1}{n} - \frac{4\hat{K}t}{c} )\hat{\omega}$ for all $t\leq \frac{c}{4n\hat{K}}$ 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 $\omega_0\geq c \eta$ on $M$ for some $c>0$ and $\phi_0$ is bounded on $M$, we construct a smooth solution to K\"ahler-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 K\"ahler-Ricci flow (see Theorem 1.1).