Holomorphic conformal structures and uniformization of complex surfaces
type: Item
instance of: scholarly article
title: Holomorphic conformal structures and uniformization of complex surfaces
zbMATH Open document ID: 0611.32023
DOI: 10.1007/BF01456283
author: Ryoichi Kobayashi
author: Isao Naruki
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164336
review text: We introduce the notion of generalized holomorphic conformal structure (GHCS) on a smooth compact complex surface and discuss the uniformization problem of surfaces admitting a GHCS. A GHCS defined by a holomorphic line bundle L on a compact complex surface X is a primitive section of \(L\otimes T^ *X\). For example, by taking \(X=P_ 2({\mathbb{C}})\) and \(L={\mathcal O}(m+3)\), we fix the way to compute the orbifold holomorphic conformal structure on the double plane branched over a curve of degree 2m with at worst simple singularities and Hilbert modular cusps. This enable us to compute explicitly the GHCS's on \(P_ 2({\mathbb{C}})\) corresponding to certain Hilbert modular surfaces and to construct a compact surface with simple singularities which fulfills \=c\({}_ 1^ 2=2\bar c_ 2\) (in the modified sense) but is not covered by the bidisc. The relation between singular Einstein-Kähler metrics and GHCS's is also discussed.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32G99
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14H15
zbMATH DE Number: 3987676
zbMATH Keywords: generalized holomorphic conformal structure
zbMATH Keywords: compact complex surface
zbMATH Keywords: uniformization
zbMATH Keywords: Hilbert modular surfaces
zbMATH Keywords: singular Einstein-Kähler metrics
MaRDI profile type: Publication
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cites work: Einstein-Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities
cites work: Polarized period map for generalized K3 surfaces and the moduli of Einstein metrics
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cites work: A General Schwarz Lemma for Kahler Manifolds
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description: scientific article; zbMATH DE number 3987676