The classification conjecture for surfaces on the Severi line

Let $X$ be a minimal surface of general type with maximal Albanese dimension. Its canonical model is denoted by $X_{\mathrm{can}}$. The surface $X$ is on the Severi line when

$$K_X^2=4\chi({\mathcal O}_X).$$

Classification conjecture. The surface $X$ is on the Severi line if and only if its canonical model $X_{\mathrm{can}}$ is a flat double cover of an Abelian surface.

This characterizes the equality case of the Severi inequality for surfaces of maximal Albanese dimension. The conjecture was confirmed in characteristic zero by Barja–Pardini–Stoppino and independently by Lu–Zuo.