nLab logarithmic geometry

Contents

Context

Geometry

under construction

Idea
Definition
Examples

Closed immersion of zero-loci
Affine line
Complex plane

Related concepts
References

Idea

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).

Definition

More precisely, 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 $\alpha^{-1}(R^\times) \simeq K^\times$ ;

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).

Examples

Closed immersion of zero-loci

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

D \coloneqq \{x_1 x_2 \cdots x_k = 0\} \hookrightarrow \mathbb{A}^n

of the product of $k$ variables inside the $n$ -dimensional affine space. The log-structure on $\mathbb{A}^n$ reflecting this is

\mathbb{N}^k \longrightarrow \mathcal{O}_{\mathbb{A}^n}

given by the exponential map

(n_i) \mapsto \prod_i x_i^{n_i}

(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)

Affine line

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

\alpha \colon \mathbb{N} \hookrightarrow \mathbb{Z}[\mathbb{N}] = \mathbb{Z}[t] \,.

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

\frac{1}{t} dt = d log(t)

(e.g. Pottharst, p. 4-5)

This phenomenon gives the name to logarithmic geometry.

Complex plane

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

\mathbb{R}_{\geq 0 } \times S^1 \longrightarrow \mathbb{C}

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)

Related concepts

References

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

Arthur Ogus, Lectures on logarithmic algebraic geometry, TeXed notes, 2001, pdf