Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds

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