Mathematische Annalen

, Volume 369, Issue 3–4, pp 1271–1282 | Cite as

Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds



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.


Exceptional Divisor Einstein Metrics Quotient Singularity Bisectional Curvature Birational Morphism 



We thank Chenglong Yu for discussions on material in Sect. 2, where he made a critical observation for Lemma 2.1. P. Gao and S.-T. Yau are supported by NSF Grants DMS-1308244 and PHY-0937443, and W. Zhou is partially supported by China Postdoctoral Science Foundation Grant No. 2015M571479.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.School of Mathematical SciencesTongji UniversityShanghaiChina

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