On the moduli spaces of fiber bundles of curves of genus \(\geqq 2\) (Q1592796)

Let \(X\to S\) be an analytic fiber bundle with fiber \(F\) of genus \(g\geq 2\) over a curve of genus \(g_b\). Let \(K^2=8(g_b-1) (g-1)\) and \(\chi= (g_b-1) (g-1)\). The aim of this paper is to study the subvariety parametrizing these fiber bundles in the moduli space \({\mathcal M}_{K^2,\chi}\) of surfaces of general type. The author proves the following theorem: If \(g_b> (g+1)/2\), then the connected component \({\mathcal M}\) of \({\mathcal M}_{K^2,\chi}\) containing the modulus of \(X\), parametrizes precisely the surfaces admitting analytic fiber bundle structure with the same \(g,g_b\) and \({\mathcal G}\) where \({\mathcal G}=\text{Image} (\pi_1(S) \to\Aut (F))\).