Lectures on Logarithmic Algebraic Geometry

The book under review is a thorough exposition of the foundations, basic notions, and methods of logarithmic algebraic geometry. The book gathers and systematizes results that before were dispersed in the literature in research articles or preprints.

Chapter I is an example of how thorough and systematic the book is. This chapter, more than 180 pages long, gives a detailed treatment of the category of (commutative) monoids, monoid modules and monoid algebras. Methods and results from this chapter would be useful not only in logarithmic geometry, but also in other developments, for example in toric geometry, certainly complementing the very good expositions of toric varieties in W. Fulton’s Introduction to Toric Varieties (Princeton, 1993) and D. Cox, J. Little and H. Schenk’s Toric Varieties (AMS, 2011).

In true algebraic-geometry fashion, Chapter II takes the algebra introduced in Chapter I and replaces it with the geometry underlying it. Starting with the prime spectrum of a commutative monoid, with the Zariski topology, and (pre)sheaves of monoids to construct locally monoidal spaces and monoschemes, develops the (monoidal) machinery totally analogous to the now classical Grothendieck theory of schemes. A beautiful turn of events and language takes us now to the analog of blowing up for schemes, which is appropriately a monoidal transformation for monoschemes. This chapter ends with the definition and properties of charts for morphisms.

With the foundations carefully laid out in the first two chapters, the remaining part of the book is now devoted to logarithmic geometry proper. A logarithmic structure on a scheme \(X\) is a morphism of sheaves of commutative monoids with codomain the structure sheaf of \(X\) and which induces an isomorphism on the sheaves of monoidal units. The flexibility of this definition is what gives logarithmic algebraic geometry part of its power and difficulty.

Chapter III develops the whole machinery of log schemes and morphims between them: fibered products and various types of morphisms, e.g., integral, inseparable or saturated. Chapter IV is devoted to the local part: logarithmic derivations and differentials, deformations, smoothness and flatness. Chapter V focuses on the geometry of varieties over the field of complex numbers by first constructing a Betti realization \(X_{\text{log}}\) of a log scheme \(X\) over \({\mathbb C}\) together with a proper map from \(X_{\text{log}}\) to the analytic realization \(X_{\text{an}}\) of \(X\), that is, the set of complex valued points of \(X\) with the classical topology induced by \({\mathbb C}\).

Then the task is to translate the topological and geometric properties of \(X_{\text{an}}\) to \(X_{\text{log}}\). For example, to translate the usual Betti and de Rham cohomology for \(X_{\text{an}}\) to the logarithmic \(X_{\text{log}}\) setting. This is carefully done in this chapter and we find several versions of the logarithmic Poincaré lemma showing that the analytic de Rham cohomology calculates the Betti cohomology of \(X_{\text{log}}\), as one would expect. There are several other such translations, which makes this chapter a highlight. At the same time this serves to ground the theory, fulfilling its aim to translate to the algebraic geometry setting a problem that in differential topology corresponds to compactifying a differential manifold by working with a manifold with boundary.

Any reader interested in this monograph will certainly have a background on algebraic geometry on the level of Hartshorne’s Algebraic Geometry textbook. With this in mind, a sense of déjà-vu would be almost inevitable, even from the table of contents. Hartshorne’s book is to Grothendieck’s EGA as Ogus’s monograph is to Gabber and Ramero’s still growing Foundations for Almost Ring Theory. The choice of topics in the book under review makes the book readable and self-contained, and it may become the standard reference on logarithmic geometry.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.