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 Kähler 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, $$\sum_{i=1}^k\frac{dz_i\wedge d\bar z_i}{|z_i|^2}+\sum_{i=k+1}^n dz_i\wedge d\bar z_i$$
If $D$ be a simple normal crossing divisor on Kähler variety $X$ with some type of singularities, like conic, cusp,...., then the Kähler 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 $\lambda=0, -1$
Reference: Paper of Unnormalize conical Kähler-Ricci flow Liangming Shen http://arxiv.org/abs/1411.7284
- $C^{2,α}$-estimate for conical Kähler-Ricci flow http://arxiv.org/abs/1412.2420