If $D$ be a simple normal crossing divisor on Kaehler variety $X$ with some type of singularities, like conic, cusp,...., then the Kahler Ricci flow

$$\frac{\partial \omega}{\partial t}=-Ric(\omega)+\lambda\omega$$

 has solution on non-compact and quasi projective, open variety $X\setminus D$ when $\lambda=0, -1$



Definition of conic metric on complement of divisor: If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a type of singularities normal to each component of $D$; in local coordinates, if $D = (z_1,...,z_k)$, the conic metric $\omega$ is  quasi-isometric with the following model, $$\sum_{i=1}^k\frac{dz_i\wedge d\bar z_i}{|z_i|^2}+\sum_{i=k+1}^n dz_i\wedge d\bar z_i$$