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

Gurjar, Rajendra Vasant; Purnaprajna, Bangere P.

J. Math. Soc. Japan 70, No. 3, 953-974 (2018).

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

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