On the Shafarevich conjecture for genus-2 fibrations (Q1006802)

Let \(X\) be a smooth projective minimal surface of general type, let \(\pi : X \rightarrow C\) be a genus two fibration over a smooth projective curve and assume \(K_X\) relatively ample. In this paper it is proved that, if moreover \(\pi\) is not a \(C^{\infty}\)-locally trivial fibre bundle, the fundamental group of a general fibre maps trivially on \(\pi_1(X)\). An analogous result for elliptic surfaces was proved by \textit{R. V. Gurjar} and \textit{A. R. Shastri} [Compos. Math. 54, 95--104 (1985; Zbl 0583.14013)]. As a consequence, the authors prove that the Shafarevich conjecture is true for \(X\) (as in [loc. cit.] for the elliptic surfaces). Further corollaries are that \(\pi_2(X)\) is a free abelian group and the proof of a conjecture of \textit{M. V. Nori} [Ann. Sci. Éc. Norm. Super., IV. Sér. 16, 305--344 (1983; Zbl 0527.14016)].