Surfaces of Albanese general type and the Severi conjecture

In 1932, F. Severi claimed, with an incorrect proof, that every smooth minimal projective surface S of irregularity q = q ( S ) > 0 without irrational pencils of genus q satisfies the topological inequality 2 c 2 1 ( S ) ≥ c 2 ( S ). According to the Enriques‐Kodaira's 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's conjecture. In this paper we prove Severi's conjecture under the additional mild hypothesis that S has ample canonical bundle. Moreover, under the same assumption, we prove that 2 c 2 1 ( S ) = c 2 ( S ) if and only if S is a double cover of an abelian surface. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)