The fundamental group of Galois cover of the surface \(\mathbb{T}\times\mathbb{T}\). (English) Zbl 1156.14310
This is the final paper in a series of four [Acta Appl. Math. 75, No. 1–3, 195–270 (2003; Zbl 1085.14504); Osaka J. Math. 40, No.4, 857–893 (2003; Zbl 1080.14516); Commun. Algebra 34, No.1, 89–106 (2006; Zbl 1086.14014)], concerning the surface \(T \times T\) embedded in \(\mathbb{CP}^8\), where \(T\) is a the one dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto \(\mathbb{CP}^2\), and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.
MSC:
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
| 14H30 | Coverings of curves, fundamental group |
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