Nonexistence for complete K\"ahler Einstein metrics on some noncompact manifolds

Abstract

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.