Let $(X,\omega)$ be a complete noncompact Kahler manifold of finite volume. Suppose $X$ is can be compactified to a compact projective manifold $M$ so that $D=M-X$ is a divisor of simple normal crossings. My question is:
Is the Kahler metric $\omega$ on $X$ extendable (in certain natural way) to the compactification $M$?
When such extensions exist, what can one say about such metric? For example, volume, curvature, etc.