A simple proof of Bogomolov's theorem on class \(\nabla II_ 0\) surfaces with \(b_ 2=0\)
type: Item
instance of: scholarly article
title: A simple proof of Bogomolov's theorem on class \(\nabla II_ 0\) surfaces with \(b_ 2=0\)
zbMATH Open document ID: 0705.32018
author: Jun Li
author: Shing-Tung Yau
author: Fangyang Zheng
published in: Illinois Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Kodaira has almost completed the classification of compact complex analytic surfaces. The only class of surfaces whose structure remained unclear was class \(VII_ 0\)= \{minimal surfaces with \(b_ 1=0\}\) (where \(b_ k\) denotes the k-th Betti number). Kodaira has shown that if the algebraic dimension (the transcendence degree of the field of meromorphic functions) a(S) of a surface S from class \(VII_ 0\) is equal to one (of course, for surfaces of class \(VII_ 0\) a(S)\(\leq 1)\), then S is an elliptic surface, and if \(a(S)=0\), \(b_ 2(S)=0\) and S contains curves, then S is a Hopf surface (i.e. a surface whose universal cover \({\mathbb{C}}^ 2\setminus \{(0,0)\})\). Thus the only two subclasses that had to be classified are as follows: \((A)=\{S\in VII_ 0|\) \(b_ 2(S)=0,\) S does not contain curves\}; \((B)=\{S\in VII_ 0|\) \(b_ 2(S)>0\}.\) The structure of class (B) is not yet completely clear. As for (A), Inoue constructed three types of examples, viz \(S_ M\), \(S_ t^{(+)}\), \(S^-\) parameterized by a matrix M, a scalar t and some integers. They all have the form \(H\times {\mathbb{C}}/G\), where H is the upper half plane, G is a discrete subgroup of the affine group acting freely and properly discontinuously on \(H\times {\mathbb{C}}\). Inoue also proved that if \(S\in (A)\) and there exists a bundle L on S such that \(H^ 0(S,T_ S\otimes L)\neq 0\), where \(T_ S\) is the tangent bundle of S, then S is of one of the above three types. Hence the remaining surfaces in (A) belong to \[ (A_ 0)=\{S\in (A)| \quad H^ 0(S,T_ S\otimes L)=0,\quad \forall L\in Pic\;S\}. \] Finally, \textit{F. A. Bogomolov} proved that \((A_ 0)=0\) [cf. Math. USSR, Izv. 10(1976), 255- 269 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 273-288 (1976; Zbl 0352.32020); Math. USSR, Izv. 21(1982), 31-73 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 710-761 (1982; Zbl 0527.14029)]. His proof is complicated and hard to follow. In the note under review the authors g…
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32L20
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 81T13
zbMATH DE Number: 4156908
zbMATH Keywords: surface of class \(VII_ 0\)
zbMATH Keywords: holomorphic affine structure
zbMATH Keywords: \(\mu \) -stable bundle
zbMATH Keywords: Yang-Mills connection
zbMATH Keywords: classification of compact complex analytic surfaces
reviewed by: F. L. Zak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4156908