Metric on the complement of a divisor is complete?

Question

Let $\bar{M}$ be a projective manifold, $D$ a smooth ample divisor with normal crossings, and $S$ a defining section of $D$. It is claimed that the metric on $\bar{M} \setminus D$ corresponding to the Kähler form $$\frac{i}{2\pi} \partial \bar{\partial} (-\text{log}\Vert S \Vert^2)^{\frac{n+1}{n}}$$ is complete. How does one see this?