Global structure of an elliptic fibration.
In length and style, this paper is very much like a monograph. The author of this clearly structured and self-contained exposition has been an expert on elliptic fibrations for more than 15 years. Kodaira's theory of elliptic surfaces, which dates back to the 1960's, is a landmark in the study of elliptic fibrations. However, the higher-dimensional analytic case is more difficult and less clear. Recently, the local structure of projective elliptic fibrations was studied by \textit{N. Nakayama} [in: Higher dimensional birational geometry. Kyoto 1997, Adv. Stud. Pure Math. 35, 185--295 (2002; Zbl 1059.14015)]. The goal of the paper under review is a globalisation of these results. More specifically, the author aims at a description of the set of bimeromorphic equivalence classes of elliptic fibrations over a fixed base which are locally bimeromorphic to a projective morphism and which are equipped with a fixed marking. In addition, a reduced effective divisor in the base is fixed such that the open immersion of its complement is a toroidal embedding. All the fibrations are supposed to be bimeromorphically equivalent to a smooth fibration over this open complement. In the algebraic category, such equivalence classes are described by the Tate-Shafarevich group, which has an interpretation as an étale cohomology group. But, due to the presence of logarithmic transformations which produce multiple fibres and varieties which are not bimeromorphically equivalent, this is not sufficient in the complex analytic case. In order to treat multiple fibres, the author introduces the category of so-called \(\partial\)-spaces and equips it with a Grothendieck topology called the \(\partial\)-étale topology. If \(X\) is a normal complex variety and \(B\) a reduced Weil divisor on \(X\), it is interesting to observe that a reflexive sheaf on the \(\partial\)-space, which is determined by \((X,B)\), can be seen as a parabolic sheaf on \(X\). An important role in the study of elliptic fibrations is played by the so-called basic elliptic fibrations, which are those admitting a meromorphic section. The first \(\partial\)-étale cohomology group of the \(\partial\)-étale sheaf of germs of meromorphic sections of a fixed basic elliptic fibration plays the role of the Tate-Shafarevich group, but it also contains more information. The main result of this work states that the set of bimeromorphic equivalence classes, described above, injects as a subgroup into this cohomology group. Furthermore, the torsion part of it corresponds precisely to those fibrations which are projective. In addition, methods to compute this cohomology group via Čech cohomology and a Leray spectral sequence are developed. The final quarter of this paper is devoted to interesting applications. In particular, results of Dolgachev and Gross about elliptic threefolds are generalised to higher dimensions. Moreover, projectivity of elliptic surfaces is studied and quite a few other interesting applications are presented.
scientific article; zbMATH DE number 1833283