An orbifold approach to Severi Inequality

For a smooth minimal surface of general type $S$ with $Albdim(S) = 2$, Severi inequality says that $K_S^2 \geq 4\chi(S)$, which was proved by Pardini. It is expected that when the equality is attained, $S$ is birational to a double cover over an Abelian surface branched along a divisor having at most negligible singularities. This was proved when $K_S$ is ample by Manetti. In this paper, we applied Manetti's method to the canonical model of $S$, with some additional assumptions we proved Severi inequality and characterized the surfaces with $K_S^2 = 4\chi(S)$.In addition, we gave a characterization of the double cover over an Abelian surface via the ramification divisor.