  Let $M$ be a compact K\"ahler manifold and $N$ be a subvariety with
codimension greater than or equal to 2. We show that there are no complete
K\"ahler--Einstein metrics on $M-N$. As an application, let $E$ be an
exceptional divisor of $M$. Then $M-E$ cannot admit any complete
K\"ahler--Einstein metric if blow-down of $E$ is a complex variety with only
canonical or terminal singularities. A similar result is shown for pairs.
