Surfaces of Albanese General Type and the Severi Conjecture

In 1932 F. Severi claimed, with an incorrect proof, that every smooth minimal pro-jective surface S such that the bundle Ω 1 S is generically generated by global sections satisfies the topological inequality 2c 2 1 (S) ≥ c2(S). According to Enriques-Kodaira classification, the above inequality is easily verified when the Kodaira dimension of the surface is ≤ 1, while for surfaces of general type it is still an open problem known as Severi conjecture. In this paper we prove Severi conjecture under the additional mild hypothesis that S has ample canonical bundle. Moreover, under the same assumption, we prove that 2c 2 1 (S) = c2(S) if and only if S is a double cover of an abelian surface.