The geography of irregular surfaces (Q2918465)

The purpose of this nice article is to give a broad overview of some recent developments in the study of irregular complex surfaces of general type. The first section gives a short introduction to aims and motivations of the paper. In the second section the authors report on results and problems regarding the geography of minimal irregular surface of general type. Furthermore, they give several interesting examples. In Section 3 they briefly report on the Castelnuovo-de Franchis inequality and on a recent generalization to the case of Kähler manifolds. Section 4 is about the slope inequality, its history e proofs. Finally, Section 5 is about the Severi inequality which says that \(K^2_S \geq 4\chi(S)\) for a minimal surfaces of general type \(S\) with Albdim\((S)=2\). After many special proofs, the inequality was proved by Pardini. They give a sketch of the Manetti's proof for surfaces of general type whose canonical bundle is ample and of the elegant Pardini's proof. There are many recent interesting results in this area and the authors focus on refinements, open questions and recent generalizations. Most of them have been proved by the authors of this article. The article has an exhaustive bibliography.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].