Monatshefte für Mathematik

, Volume 174, Issue 4, pp 599–616 | Cite as

On Kähler metrisability of two-dimensional complex projective structures



We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Furthermore we show that the (pseudo-)Kähler metrics defined on some domain in the projective plane which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms on \(\mathbb {C}^3\) whose rank is at least two. This is achieved by prolonging the relevant finite-type first order linear differential system to closed form. Along the way we derive the complex projective Weyl and Liouville curvature using the language of Cartan geometries.


Complex projective geometry Cartan geometry Metrisability 

Mathematics Subject Classification (2010)

53A20 53B10 


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Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZurichSwitzerland

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