under construction
Logarithmic geometry is a slight variant of algebraic geometry (resp. analytic geometry) where schemes (resp. analytic spaces) and morphisms with mild (“logarithmic”) singularities still behave as smooth schemes (resp. smooth analytic spaces).
Where an affine variety is the formal dual to a commutative ring $(R,\times, +)$, the analog in logarithmic geometry is such a ring equipped with
a monoid $K$ and a monoid homomorphism $\alpha \colon K \longrightarrow (R, \times)$;
such that $\alpha^{-1}(R^\times) \simeq R^\times$;
where $R^\times$ is the group of units of $R$. These two items together are called a log-structure on $R$ (or a pre-log structure if the condition in the second item does not necessarily hold).
The archetypical example of a logarithmic structure, which gives the concept its name,is that describing logarithmic singularities at closed immersions which are locally of the form of the zero locus
of the product of $k$ variables inside the $n$-dimensional affine space. The log-structure on $\mathbb{A}^n$ reflecting this is
given by the exponential map
(e.g. Pottharst, p. 4)
The definition of Kähler differential forms in logarithmic geometry is such that the differential 1-forms on the corresponding log scheme here are generated over $\mathcal{O}_{\mathbb{A}^n}$ by the ordinary differentials $d x^i$ for $k \lt i \leq n$ and by the differential forms with logarithmic singularities at $D$ which are $\frac{1}{x^i} d x^i = d log x^i$ for $1 \leq i \leq k$.
(e.g. Pottharst, p. 5)
Given the affine line $\mathbb{A}^1$ with function ring $\mathcal{O}(\mathbb{A}^1) = \mathbb{Z}[t]$ then there is a log structure given by the canonical map
The sheaf of differential forms on the resulting log-scheme is that of the ordinary affine line and one more generating section which is the differential form with logarithmic singularities
(e.g. Pottharst, p. 4-5)
This phenomenon gives the name to logarithmic geometry.
Working over $\mathbb{C}$ consider the multiplicative sub-monoid $\mathbb{R}_{\geq 0} \times S^1 \subset \mathbb{C} \times \mathbb{C}$ and the map
given by product operation. This defines a log-structure on $\mathbb{C}$. The corresponding log-space is often denoted $T$ (e.g. Kato-Nakayama 99, p. 5, Ogus01, section 3.1).
Given any log-scheme $X$ over $\mathbb{C}$, then the underlying topological space $X^{log}$ has as points the log-scheme homomorphisms $T \longrightarrow X$. (Ogus 01, def. 3.1.1)
Logarithmic geometry originates with Fontaine and Luc Illusie in the 1980s.
Kazuya Kato, Logarithmic structures of Fontaine-Illusie, in Algebraic Analysis, Geometry and Number Theory, The Johns Hopkins University Press (1989), 191-224.
Luc Illusie, Logarithmic spaces (according to Kato), in: Barsotti-Symposium in Algebraic Geometry (ed. V. Cristante and William Messing), Academic Press: Perspectives in Math. 15 (1994), 183-203
Luc Illusie, Arthur Ogus, Géométrie logarithmique (pdf)
Luc Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic ́etale cohomology. Astérisque, (279):271–322, 2002. Cohomologies p-adiques et applications arithmétiques, II.
Kazuya Kato, Chikara Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over ${\bf C}$, Kodai Math. J. Volume 22, Number 2 (1999), 161-186 (Euclid)
Brief surveys include
Pottharst, Logarithmic structures on schemes (pdf)
Arthur Ogus, Logarithmic geometry, talk slides 2009 (pdf)
Dan Abramovich, Logarithmic geometry and moduli, talk slides 2014 (pdf)
Lecture notes include
See also
Dan Abramovich, Qile Chen, Danny Gillam, Yuhao Huang, Martin Olsson, Matthew Satriano, Shenghao Sun, Logarithmic geometry and moduli, arxiv/1006.5870
Martin C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. Ecole Norm. Sup. (4), 36(5):747{791, 2003.
Jakob Stix, section 3 of Projective Anabelian Curves in Positive Characteristic and Descent Theory for Log-Etale Covers, 2002 (pdf)
Discussion in the context of higher algebra (brave new algebra) is in