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logarithmic geometry

under construction

Contents

Idea

Logarithmic algebraic geometry (or logarithmic geometry for short) is a slight variant of algebraic geometry where schemes and morphisms with mild (“logarithmic”) singularities still behave as smooth schemes.

Where an affine variety is the formal dual to a commutative ring (R,×,+)(R,\times, +), the analog in logarithmic geometriy 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 July 1, 2014 02:47:13 by Urs Schreiber (89.204.138.13)