Holomorphic conformal structures and uniformization of complex surfaces
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.
generalized holomorphic conformal structure
scientific article; zbMATH DE number 3987676