##
**On the fundamental group of a smooth projective surface with a finite group of automorphisms.**
*(English)*
Zbl 1407.14015

The authors consider quotients \(X/G\) of smooth projective surfaces \(X\) by the action of a finite group. More precisely they study the relation between \(\pi_1(X)\) and \(\pi_1(X/G)\). In their main theorem it is assumed that \(X/G\) admits a minimal \(\mathbb P^1\) fibration \(X/G \to C\). Stein factorization applied to \(X \to C\) yields a fibration \(X \to D\) onto a smooth curve \(D\) with general fiber \(F\). The statement of the theorem is that the induced homomorphism \(\pi_1(F) \to \pi_1(X)\) is finite under certain technical assumptions on \(G\) and \(X\). In particular \(\pi_1(X) \to \pi_1(X/G)\) has then finite kernel and cokernel. As a corollary they deduce that Shavarevich’s conjecture is valid for such surfaces i.e. the universal covering space is holomorphically convex and answer a question of Nori about fundamental groups and free abelianness of second homotopy groups for these surfaces. Moreover they prove a technical result, which can be useful in other situations, about multiplicities of fibres of fibrations with a finite group of automorphisms.

Reviewer: Christian Gleissner (Bayreuth)

### MSC:

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

14J29 | Surfaces of general type |

14L30 | Group actions on varieties or schemes (quotients) |

14H30 | Coverings of curves, fundamental group |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |

14J50 | Automorphisms of surfaces and higher-dimensional varieties |

### Keywords:

surfaces of general type; fibrations; fundamental groups of algebraic surfaces; Shafarevich conjecture; holomorphic convexity; finite group actions on varieties; base change
PDF
BibTeX
XML
Cite

\textit{R. V. Gurjar} and \textit{B. P. Purnaprajna}, J. Math. Soc. Japan 70, No. 3, 953--974 (2018; Zbl 1407.14015)

Full Text:
DOI

### References:

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.