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 2c(1)(2) (S) greater than or equal to c(2) (S). According to the Enriques-Kodaira's classification, the above inequality is easily verified when the Kodaira dimension of the surface is less than or equal to 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 2c(1)(2)(S) = c(2) (S) if and only if S is a double cover of an abelian surface. (C) 2003 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim.