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 → C of genus g.We prove a linear upper bound on the genus g if KS²≤ 4χ(OS). Examples are constructed showing that the above linear upper bound is sharp.We also construct a sequence of surfaces Sₙ of general type with KSₙ²/χ(OSₙ)>4 and with an Albanese fibration fₙ, such that the genus gₙ of a general fiber of fₙ increases quadratically with χ(OSₙ), and that KSₙ²/χ(OSₙ) can be arbitrarily close to $4$.