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Simply-connected algebraic surfaces of positive index. (English) Zbl 0627.14019
From the introduction: “In this paper we construct simply-connected algebraic surfaces of general type which have positive index. After the famous inequality $$c^ 2_ 1\leq 3c_ 2$$ [Bogomolov-Miyaoka-Yau (1976)] was discovered, the empirical evidence for simply-connected surfaces of general type strongly indicated that in their case the inequality $$c^ 2_ 1\leq 2c_ 2$$ must hold. The inequality $$c^ 2_ 1\leq 2c_ 2$$ is equivalent of course to the fact that the index $$\tau (=(c^ 2_ 1- 2c_ 2)/3)$$ is not positive. So the following conjecture was formulated [“watershed conjecture of Bogomolov”: cf. J.-M. Feustel and R.-P. Holzapfel, Math. Nachr. 111, 7-40 (1983; Zbl 0528.14015)]: If Y is a surface of general type with $$\tau (Y)>0$$ then $$\pi_ 1(Y)\neq 0$$ (or equivalently $$\pi_ 1(Y)$$ is infinite).
Thus our results provide counter-examples to this conjecture. Take $$X={\mathbb{C}}P^ 1\times {\mathbb{C}}P^ 1$$. Let $$\ell_ i\subseteq X$$, $$\ell_ 1\subseteq {\mathbb{C}}P^ 1\times pt$$, $$\ell_ 2=pt\otimes {\mathbb{C}}P^ 1$$. Let $$E=a\ell_ 1+b\ell_ 2$$, a,b$$\in {\mathbb{N}}$$. Let $$X_{a,b}$$ be the embedding of X into $${\mathbb{C}}P^ N$$ with respect to the linear system $$| E|$$. Take a canonical projection f of $$X_{a,b}$$ to $${\mathbb{C}}P^ 2$$, $$n=\deg (f)=2ab$$. Let Y be its Galois cover that corresponds to the full symmetric group $$S_ n$$. For $$a\gg 0$$, $$b\gg 0$$, a and b relatively prime, Y is the example we present. In our work we prove that the fundamental group $$\pi_ 1(Y)$$ is a finite abelian group and that it is trivial when a,b are relatively prime.”
Irrelevant reviewer’s remark: some names of authors are not correct, e.g. Bogamolov instead of Bogomolov, twice in the introduction, Holzupfel instead of Holzapfel in the references.
Reviewer: E.Stagnaro

##### MSC:
 14F45 Topological properties in algebraic geometry 14F35 Homotopy theory and fundamental groups in algebraic geometry 57R20 Characteristic classes and numbers in differential topology 14E20 Coverings in algebraic geometry 14J25 Special surfaces
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##### References:
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