Rigid geometry and applications

Abstract

In this paper we present a survey of rigid geometry. Here, special emphasis is put on the so-called "birational approach" to rigid geometry, which adopts classical methods of birational geometry to the theory of rigid spaces. The paper is divided into three parts. Part I is a general introduction to rigid geometry a la J. Tate and M. Raynaud. In Part II we are to overview the birational approach to rigid geometry, which combines the idea of Raynaud and that of O. Zariski, as one of the conceptual starting points of rigid geometry. In Part III we discuss some applications, which reveal the effectiveness of the ideas in rigid geometry that arise from our viewpoint.