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logarithmic geometry

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,×,+)(R,\times, +), the analog in logarithmic geometry is such a ring equipped with

  1. a monoid KK and a monoid homomorphism α:K(R,×)\alpha \colon K \longrightarrow (R, \times);

  2. such that α 1(R ×)K ×\alpha^{-1}(R^\times) \simeq K^\times;

where R ×R^\times is the group of units of RR. These two items together are called a log-structure on RR (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{x 1x 2x k=0}𝔸 n D \coloneqq \{x_1 x_2 \cdots x_k = 0\} \hookrightarrow \mathbb{A}^n

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

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

given by the exponential map

(n i) ix i n i (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 𝒪 𝔸 n\mathcal{O}_{\mathbb{A}^n} by the ordinary differentials dx id x^i for k<ink \lt i \leq n and by the differential forms with logarithmic singularities at DD which are 1x idx i=dlogx i\frac{1}{x^i} d x^i = d log x^i for 1ik1 \leq i \leq k.

(e.g. Pottharst, p. 5)

Affine line

Given the affine line 𝔸 1\mathbb{A}^1 with function ring 𝒪(𝔸 1)=[t]\mathcal{O}(\mathbb{A}^1) = \mathbb{Z}[t] then there is a log structure given by the canonical map

α:[]=[t]. \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

1tdt=dlog(t) \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 0×S 1×\mathbb{R}_{\geq 0} \times S^1 \subset \mathbb{C} \times \mathbb{C} and the map

0×S 1 \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 TT (e.g. Kato-Nakayama 99, p. 5, Ogus01, section 3.1).

Given any log-scheme XX over \mathbb{C}, then the underlying topological space X logX^{log} has as points the log-scheme homomorphisms TXT \longrightarrow X. (Ogus 01, def. 3.1.1)

References

Logarithmic geometry originates with Fontaine and Luc Illusie in the 1980s.

Brief surveys include

Lecture notes include

See also

The role of log geometry in motivic integration is studied in

Discussion in the context of higher algebra (brave new algebra) is in