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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).

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 ×)R ×\alpha^{-1}(R^\times) \simeq R^\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).

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)

Examples

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.

  • 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 bfC{\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

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

Revised on November 14, 2014 10:23:31 by Frédéric Paugam (134.157.13.171)