Geometry & Topology

Polyhedral Kähler manifolds

Dmitri Panov

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800291

Digital Object Identifier
doi:10.2140/gt.2009.13.2205

Mathematical Reviews number (MathSciNet)
MR2507118

Zentralblatt MATH identifier
1175.53082

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

Keywords
polyhedral metric Kobayashi–Hitchin correspondence line arrangement

Citation

Panov, Dmitri. Polyhedral Kähler manifolds. Geom. Topol. 13 (2009), no. 4, 2205--2252. doi:10.2140/gt.2009.13.2205. https://projecteuclid.org/euclid.gt/1513800291


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