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A simple proof of Bogomolov’s theorem on class \(\nabla II_ 0\) surfaces with \(b_ 2=0\). (English) Zbl 0705.32018
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, 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 give a short proof of Bogomolov’s theorem based on the vanishing theorem of M. Inoue, Sh. Kobayashi and T. Ochiai [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 247-264 (1980; Zbl 0467.32014)] and the stability criterion of the first two authors [in: Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 560-573 (1987; Zbl 0664.53011)], and the arithmetical algebraic and geometric properties of these rational surfaces. To this end, the authors study the action of the Galois group \(Gal(\bar k/k)\) on \(Pic\bar X,\) where \(\bar k\) is the separable closure of k and \(\bar X=X_{Spec}\otimes Spec \bar k\). The techniques used in the paper include Weyl groups and root systems, Galois cohomology, algebraic tori in semi-simple groups and conic bundles. The paper contains a lot of explicit examples.
Reviewer: F.L.Zak

MSC:
32J15 Compact complex surfaces
32L20 Vanishing theorems
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53C05 Connections, general theory
14J25 Special surfaces
81T13 Yang-Mills and other gauge theories in quantum field theory
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