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 \(2c_ 2=c^ 2_ 1\).

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) |