Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds

Abstract

Let M be a compact Kähler manifold and N be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler-Einstein metrics on M − N . As an application, let E be an exceptional divisor of M . Then M − E cannot admit any complete Kähler-Einstein metric if blow-down of E is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs. Problem 1.1. Let M be a compact Kähler manifold and N be a subvariety with codimension bigger than or equal to 2, how to find a complete canonical metric on the noncompact Kähler manifold M − N ? Date: March 22, 2016.

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

“…Remark 1.3. By Gao-Yau-Zhou [19], if M is a compact Kähler manifold and N is a subvariety with codimension greater than or equal to 2, then there are no complete Kähler-Einstein metrics on M − N . Corollary 1.5 shows that this conclusion does not hold for the conformally Kähler Einstein metric.…”

“…we have ϕ(x 0 ) = ϕ ′ (x 0 ) = 0. Then by (2.2), we obtain γ = µ(ν+λx 0 ) (b+ax 0 ) 2 < 0, µ < 0 and 19), we get ϕ(x 1 ) > 0. This is in contradiction with ϕ(x 1 ) = 0.…”

Landau–Ginzburg models for certain fiber products with curves

“…One set of examples is discussed in[4]. A second set of examples arises by removing an anticanonical divisor with deep singularities from a projective manifold, then the complement will not have a complete Ricci-flat metric, though it may still have a non-complete Ricci-flat metric.…”