Geometriae Dedicata

, Volume 138, Issue 1, pp 151–172 | Cite as

Compact complex surfaces and constant scalar curvature Kähler metrics

  • Yujen Shu
Original Paper


In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of \({\mathbb {P}_2}\) at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kähler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.


Deformation equivalence Constant scalar curvature Kähler metric 

Mathematics Subject Classification (2000)

53C25 53C55 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

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