The minimal volume of log surfaces of general type with positive geometric genus

Abstract

We determine the minimal possible volume of a projective log canonical surface of general type with prescribed positive geometric genus. As applications, we provide effecitive Noether type inequalities for log canonical threefolds and stable surfaces. Also, we obtain a uniform bound on the order of the symplectic automorphism group $\mathrm{Aut}_s(S)$ of smooth projective surfaces $S$ of general type.