Geometry & Topology

Polyhedral Kähler manifolds

Dmitri Panov

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In this article we introduce the notion of polyhedral Kähler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4–dimensional case, prove that such manifolds are smooth complex surfaces and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. A parabolic version of the Kobayshi–Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kähler metrics of nonnegative curvature on P2 with singularities at complex line arrangements.

Article information

Geom. Topol., Volume 13, Number 4 (2009), 2205-2252.

Received: 29 January 2009
Revised: 5 May 2009
Accepted: 26 April 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C56: Other complex differential geometry [See also 32Cxx]
Secondary: 32Q15: Kähler manifolds 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

polyhedral metric Kobayashi–Hitchin correspondence line arrangement


Panov, Dmitri. Polyhedral Kähler manifolds. Geom. Topol. 13 (2009), no. 4, 2205--2252. doi:10.2140/gt.2009.13.2205.

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