Global Structure of an Elliptic Fibration

Abstract

∂-étale topology is introduced for analytic spaces with boundary as an analog of étale topology for schemes. A locally projective elliptic fibration is bimeromorphically considered as a torsor in the ∂-étale topology of the associated basic elliptic fibration. The related ∂-étale cohomology groups have much information on the structure of elliptic fibrations. In particular, an answer to Ueno's extension problem, a relation to Tate–Shafarevich groups and their finiteness properties, characterizations of projective and Kähler elliptic fibrations, and a generalization of logarithmic transformation to arbitrary dimension are obtained. This article is a revised version of [N5].