# nLab logarithmic geometry

Contents

### Context

#### Geometry

Ingredients

Concepts

Constructions

Examples

Theorems

under construction

# Contents

## 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

1. a monoid $K$ and a monoid homomorphism $\alpha \colon K \longrightarrow (R, \times)$;

2. 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)

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