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

Let $\bar{M}$ be a compact complex manifold with smooth K"ahler metric $\eta$, and let $D$ be a smooth divisor on $\bar{M}$. Let $M=\bar{M}\setminus D$ and let $\widehat{\omega }$ be a Carlson-Griffiths type metric on $M$. We study complete solutions to K"ahler-Ricci flow on $M$ which are comparable to $\widehat{\omega }$, starting from a smooth initial metric ${\omega }_0=\eta +i\mathrm{\partial }\bar{\mathrm{\partial }}{\varphi }_0$ where $\phi_0\in C^{\infty}(M)$\mathrm{.}When$\omega_0\geq c \hat{\omega}$on$M$forsome$c>0$and$\phi_0$haszeroLelongnumber,weconstructasmoothsolution$\omega(t)$ 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 }$sothat$\omega(t)\geq (\frac{1}{n} - \frac{4\hat{K}t}{c} )\hat{\omega}$forall$t\leq \frac{c}{4n\hat{K}}$where$\hat{K}$ is a non-negative upper bound on the bisectional curvatures of $\widehat{\omega }$ (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 $\widehat{\omega }$ 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 K"ahler-Ricci on $M\times [0,T_{[{\omega }_0]})$ which is equivalent to $\widehat{\omega }$ for all positive times. This includes as a special case when ${\omega }_0$ is smooth on $\bar{M}$ in which case the solution becomes instantaneously complete on $M$ under K"ahler-Ricci flow (see Theorem 1.1).