As a variant of algebraic geometry, the objects of study in logarithmic geometry are schemes with additional structures, so-called logarithmic structures, or log structures for short. These structures typically contain the boundary or infinitesimal information of schemes, therefore, logarithmic geometry provides a geometric framework to investigate two fundamental problems in algebraic geometry: compactification and degenerations. Any scheme can be endowed with a trivial log structure and compatible with morphisms, in this sense, logarithmic geometry can be thought of as some kind of enlargement of algebraic geometry.
Logarithmic geometry was introduced by Fontain-Illusie, Deligne, and Faltings to study de Rham complexes with logarithmic poles. Its foundation was given by Kato in the late 1980s and used for studying \(p\)-adic Galois representations associated to varieties with bad reductions. This has been carried out by Kato, Hyodo, Faltings, Tsuji, and others in \(p\)-adic Hodge theory. Finally, they have succeeded to prove the semi-stable conjecture of Fontaine-Jannsen. Later it spread to other areas, especially moduli theory, as singular varieties naturally occur at the boundary of many moduli spaces . With log structures, it is reasonable to expect that we get proper moduli spaces. For example, we know that there exists a moduli space \(\mathcal{M}_{g,n}\) for \(n\)-marked points of smooth curves of genus \(g\geq 2\). By a theorem of Deligne and Mumford, this space can be compactified using semi-stable curves. Kato shows this compactification can also be interpreted as a moduli space of log smooth curves. It is natural to study moduli problems in stack theory. The relation between logarithmic geometry and stacks was studied by Illusie and Olsson. There are also have many other applications, such as Gabber’s purity result or Bloch conductor formula.
The book under review is a self-contained exposition of the foundations of logarithmic geometry and would serve as a standard textbook for graduate students on this subject. The approach is scheme-theoretic, therefore the reader only needs to have a basic knowledge of algebraic geometry. Similar to the theory of commutative rings, the theory of monoids is presented in the first chapter. More precisely, the properties of homomorphisms of monoids, such as exactness, are studied in detail. Chapter II discusses sheaves of monoids on topological spaces, especially, the notions of monoschemes, charts, and coherence. After the 270 pages long and detailed preliminaries in the first two chapters, Chapter III turns to develop the theory of logarithmic schemes both in the Zariski and étale topologies. Chapter 4 studies the local properties of morphisms, such as smoothness and flatness. The last chapter discusses the topology and cohomology of log schemes over the field of complex numbers \(\mathbb{C}\). We see that the Betti realization of a log smooth scheme over \(\mathbb{C}\) is a topological manifold with boundary, and analytic de Rham cohomology calculates the Betti cohomology of its Betti realization.