Lectures on Logarithmic Algebraic Geometry

Advance praise: 'Logarithmic geometry is a framework tailored for studying two fundamental aspects in algebraic geometry; compactification and degeneration. It has spectacular applications to p-adic Hodge theory, ramification, etc. Written by a top researcher in the field, this book deals with the foundation of the theory. Emphasis is placed on the geometry of monoids, on which log geometry is based, and on logarithmic smoothness, a key concept of the theory. The reader will be enabled to explore the fertile field.' Takeshi Saito, University of Tokyo Advance praise: 'Logarithmic geometry was created thirty years ago in order to construct analogues in mixed characteristic of the limiting Hodge structures of the complex setting, and has since then become a powerful tool in many branches of arithmetic geometry. This long-awaited monograph presents a gentle, self-contained, and systematic exposition of the basics of the theory of log schemes (including a detailed discussion of the geometry of monoids and fans), and elegant applications to de Rham and Betti cohomology.' Luc Illusie, Universite Paris-Sud Advance praise: 'In the last three decades, logarithmic geometry has become a key tool in many areas of arithmetic and algebraic geometry (moduli problems, p-adic Hodge theory ...). Arthur Ogus' book, patiently matured and without equivalent today, provides the first systematic study of the subject. In particular, it contains a careful study of monoids and henceforth provides the invaluable references that were lacking to put the whole theory on a firm basis. It concludes with some of the possible applications showing the power of the theory.' Ahmed Abbes, Centre national de la recherche scientifique and Institut des Hautes Etudes Scientifiques