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Uniformization of complex surfaces. (English) Zbl 0755.32024
Kähler metric and moduli spaces, Adv. Stud. Pure Math. 18, No. 2, 313-394 (1990).
[For the entire collection see Zbl 0743.00065.]
Pointing out five approaches to uniformization of complex manifolds in dimension $$\geq 2$$ (a concept which is given a precise meaning), the author concentrates on two: (1) the search for numerical characterizations of orbisurfaces that are quotients of the 2-ball by a discrete group: (2) the study of generalized holomorphic conformal structures. Case (1) is related to the existence of a canonical Kähler- Einstein metric on an orbisurface $$(X,D)$$ and uses the fact that every two-dimensional log-canonical singularity is uniformizable by a bounded symmetric domain $${\mathcal D}$$ with local covering transformation group an appropriate parabolic discrete subgroup of $$\text{Aut}({\mathcal D})$$. This entails a classification of the 2-dimensional log-canonical singularities (for details of which the reader is referred to [Na], which doesn’t exist in the bibliography — is [Nak] the intended reference?). In particular, surfaces with $$3c_ 2=c_ 1^ 2$$ are characterized as ball quotients in the category of normal surfaces with branch loci. In case (2) one seeks a method of constructing a flat $$G$$-structure modelled after a standard Hermitian symmetric space $$M$$, on an orbifold that is to be uniformized. If such a flat $$G$$-structure exists that is compatible with the orbifold structure, then one can construct a differential equation satisfied by the period map of a family of algebraic varieties. One constructs so-called generalized holomorphic conformal structures: such a structure defined by a holomorphic line bundle $$L$$ on a compact surface $$X$$ is a primitive holomorphic section of $$L\otimes S^ 2T^*(X)$$. Results on the uniformization of surfaces with generalized holomorphic conformal structures have been obtained by the author and I. Naruki [Math. Ann. 279, No. 3, 485-500 (1988; Zbl 0611.32023)]. These results are applied to finding examples and counterexamples to uniformization of surfaces with $$2c_ 2=c^ 2_ 1$$.
Reviewer: J.S.Joel (Kelly)

##### MSC:
 32J15 Compact complex surfaces 32S20 Global theory of complex singularities; cohomological properties 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)