Albanese fibrations of surfaces with low slope

Let S be a minimal irregular surface of general type, whose Albanese map induces a fibration \(f:\,S \rightarrow C\) $f:S\to C$ of genus g. We prove a linear upper bound on the genus g if \(K_S^2\le 4\chi (\mathcal {O}_S)\) $K_S^2\le 4\chi ({\mathcal{O}}_S)$ , namely Examples are constructed showing that the above linear upper bound is sharp. We also give a characterization of the Albanese fibrations reaching the above upper bound when \(\chi (\mathcal {O}_S)\ge 5\) $\chi ({\mathcal{O}}_S)\ge 5$ . On the other hand, we will construct a sequence of surfaces \(S_n\) $S_n$ of general type with \(K_{S_n}^2/\chi (\mathcal {O}_{S_n})>4\) $K_{S_n}^2/\chi ({\mathcal{O}}_{S_n})>4$ and with an Albanese fibration \(f_n\) $f_n$ , such that the genus \(g_n\) $g_n$ of a general fiber of \(f_n\) $f_n$ increases quadratically with \(\chi (\mathcal {O}_{S_n})\) $\chi ({\mathcal{O}}_{S_n})$ , and that \(K_{S_n}^2/\chi (\mathcal {O}_{S_n})\) $K_{S_n}^2/\chi ({\mathcal{O}}_{S_n})$ can be arbitrarily close to 4.