# zbMATH — the first resource for mathematics

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