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I'm currently reading a thesis that uses log-structures. I should mention that this is my first encounter with them, and the thesis (as well as my expertise) is scheme-theoretic (in fact stack-theoretic) and so the original geometric motivations are lost on me.

Here is my meek understanding. For any scheme, we can give a log-structure. This is a sheaf , $M$, fibered in monoids, on the etale site over a scheme $S$; together with a morphism of sheaves fibered in moinoids $\alpha:M\rightarrow O_S$ such that when it is restricted to $\alpha^{-1}(O_S^{\times})$ it is an isomorphism.

This $\alpha$ is called the exponential map, and for any $t\in O_S(U)$ (for some $U$), a preimage of it via $\alpha$ is called $log(t)$($\in M(U)$).

I am curious about a few things, and puzzled about others. First, in terms of the notation, surely it's no coincidence that these are called exponential maps and log-structures. What is the geometric motivation for it?

Second, these come up in the thesis I'm reading in the context of tame covers. I am puzzled about what, precisely, log-structures contribute. It seems to me, in extremely vague terms (commensurate with my understanding), that the point of log-structures in this context is that if you add this extra information to tame covers it somehow helps you construct proper moduli spaces of covers.

On top of everything I'm also confused about the role of `minimal log-structures' in all of this.

In conclusion, if you can say anything at all about the motivations of log-structures in the geometric setting, or more importantly in the context of tame covers, I would extremely appreciate it. The plethora of notationally different texts on the subject is making it hard to understand the gist of what's going on.

Also, if you have examples that I should have in mind when thinking about it, that would be ideal.

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How does "sheaf fibered in monoids" differ from "sheaf of (commutative) monoids"? In the context of log structures, I have only seen the latter. –  S. Carnahan Jul 12 '11 at 3:46
    
It's the same thing. I was putting in stacky language so as to emphasize that it is a sheaf on the etale site, rather than the Zariski site. –  Makhalan Duff Jul 12 '11 at 4:05
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Pottharst has written a very clear short introduction to log-structures: math.bu.edu/people/potthars/writings/log.str.pdf –  David Hansen Jul 12 '11 at 5:37

3 Answers 3

Sometimes, it is not easy to choose a compactification of a moduli space, especially if the objects being parametrized are complicated - one may find that a choice of degenerate structure is too permissive, and is parametrized by superfluous components. One reason why log structures are useful is that they often yield parsimonious degenerations of structures (and thereby, natural compactifications of moduli spaces).

A relatively simple a posteriori example comes from the moduli of smooth pointed curves. To compactify, you can generalize smoothness to allow at most nodal singularities (Deligne-Knudsen-Mumford), or you can add a (fs) log structure, and consider the moduli problem of log-smooth integral pointed curves (F. Kato). More generally, you can compactify the moduli of smooth twisted curves using twisted stable curves (Abramovich-Vistoli), or twisted log curves (Olsson). In the case of log curves, one finds that a certain "balanced" condition appears automatically, and excludes curves whose nodes don't have matching orbifold structure.

I've never seen the map $\alpha$ referred to by names like logarithm and exponential, but the notion of logarithm is appropriate when considering differentials. If you take the affine line with the natural chart $\mathbb{N} \mapsto \mathbb{C}[\mathbb{N}] = \mathbb{C}[t]$, the sheaf of log differentials contains a section of the form $\frac{dt}{t} = d\log t$. Similarly, if you have a log-smooth curve, its sheaf of relative differentials is the dualizing sheaf, which is made of ordinary differentials on the (schematically) smooth locus, and has at most log poles at nodes, where the residues on the two pieces sum to zero.

I found the introduction of Kato-Usui (Classifying spaces of degenerating mixed Hodge structures) rather inspirational.

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Regarding tame covers: For log-schemes there are the notions of log-étale and log-smooth morphisms, which behave very similarly to the classical notions of étaleness and smoothness.

If $X\subset \overline{X}$ and $Y\subset \overline{Y}$ are open immersions of smooth $k$-schemes, for, say, $k$ a field, such that the complements of $X$ and $Y$ are strict normal crossings divisors, then $\overline{X}$ and $\overline{Y}$ get canonical (fine, saturated) log-structures. Lets call the log-schemes $X^{\log}$ and $Y^{\log}$. If $f:X\rightarrow Y$ is a finite étale morphism, extending to a finite morphism $\bar{f}:\overline{X}\rightarrow \overline{Y}$, then $f$ induces a morphism of log-schemes $f^{\log}:X^{\log}\rightarrow Y^{\log}$, and $f^{\log}$ is log-étale if and only if $\bar{f}$ is a tame covering in the usual sense. So it "behaves" like an étale covering in the category of log-schemes. For example, one can develop a theory of log-fundamengal groups and so on. A very nice reference for this is Jakob Stix thesis, which can be found on his homepage.

In fact log-étaleness is more general: For $f^{\log}$ to be log-étale, $\bar{f}$ does not have to be a finite morphism; certain non-finite ones are allowed, for example so called "log-blowups". As far as I understand they play a crucial role in developing log-étale cohomology. A good reference for this and much much more is

Illusie, Luc. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology. (English summary) Cohomologies p-adiques et applications arithmétiques, II. Astérisque No. 279 (2002), 271–322.

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I believe that people refer to $\alpha$ as the exponential map exactly because of the "big example (I)" in Pottharst's notes. Also, a small point, for any scheme we can give many log structures. For instance, in addition to the main example of a log str associated to a closed immersion, the simplest examples are $M = \mathcal O_X^*$ and $M = \mathcal O_X.$ I found K. Kato's paper "Logarithmic structures of Fontaine-Illusie" to be a good introduction.

Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 191–224, Johns Hopkins Univ. Press, Baltimore, MD, 1989.

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