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Hi everyone,

I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the moment, but I would like to have a more detailed source that goes through or gives an overview of the constructions of classical scheme theory that have analogs in the log-setup. Are there any articles/books that in your opinion are required reading if I want to learn about log-geometry? What are beautiful examples of applications of this machinery?

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  • $\begingroup$ It seems Illusie has a few articles. $\endgroup$
    – Anweshi
    Jan 21, 2010 at 21:25
  • $\begingroup$ Also look up Olsson. $\endgroup$
    – Charles Siegel
    Jan 21, 2010 at 21:26
  • $\begingroup$ there is a review of log-syntomic site in ''Torsion étale and crystalline cohomologies'' (Breuil, Messing), Astérisque 279, 2002, 81-124. But I don't know if it is what you want. I know practically nothing about log geometry... $\endgroup$
    – natura
    Jan 21, 2010 at 22:18
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    $\begingroup$ Olsson gave 5 lectures on log geometry in Pisa a few summers ago, which Anton kindly live-texed: math.berkeley.edu/~anton/written/AspectsModuli/MO.pdf $\endgroup$
    – David Zureick-Brown
    Feb 8, 2010 at 17:45
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    $\begingroup$ @etairi stacky.net/wiki/index.php?title=Course_notes $\endgroup$
    – David Zureick-Brown
    Aug 31, 2014 at 15:48

4 Answers 4

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I put up some old notes by Illusie here for you; they're very detailed and treat log smoothness, the log de Rham complex, and other topics in their second exposé. They're my favourite first reference.

There is also Ogus's book, the latest draft of which is here.

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  • $\begingroup$ Thanks a lot, the book by Ogus looks very nice, and I'll definitely have a look at Illusie's notes. $\endgroup$
    – Lars
    Jan 22, 2010 at 11:24
  • $\begingroup$ @Thanos: I tried to open the notes you mentioned, but it appeared as a broken link. Did you already delete them? I would appreciate if you can help me to reach those notes, thanks :) $\endgroup$
    – Yujia Qiu
    Jul 7, 2010 at 19:36
  • $\begingroup$ @Yujia: link reactivated! $\endgroup$
    – Thanos D. Papaïoannou
    Aug 12, 2010 at 21:17
  • $\begingroup$ The link is broken again $\endgroup$
    – David Corwin
    Oct 23, 2019 at 3:36
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    $\begingroup$ The link to Illusie's notes is broken. It's a shame -- I found them quite helpful. Were they ever in more published, citable form? $\endgroup$
    – Leo Herr
    Mar 7, 2022 at 7:47
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One standard reference is

Luc Illusie. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic ́etale cohomology. Ast ́erisque, (279):271–322, 2002. Cohomologies p-adiques et applications arithm ́etiques, II.

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I believe that Arthur Ogus has been working on a book on this topic for many years. I don't know if it (or at least some version of it) has appeared. (I looked on his web-page and found what looks like a nice set of slides from a talk, 62 pages of them, but no actual book.)

In any event, Ogus certainly has many papers on the topic. My recommendation, if you have gotten through Kato's article, would be to start reading some of Ogus's and others' articles. A lot of them are reasonably foundational, and should be accessible if you have Kato's article under your belt. In addition to the names already mentioned in the various comments and answers, Kisin has a couple of nice papers using log-schemes on his web-page.

One nice application, arithmetic in nature (and the first place that I saw log schemes), is the paper of Coleman--Voloch on companion forms. Kisin's paper on the Galois action on the prime-to-p-etale fundamental group is another nice application to arithmetic geometry that I know of.

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  • $\begingroup$ Background stuff: Ogus gave a course on this stuff in Berkeley in the 1990s and I think he was already working on the book then :-) Coleman once told me that he was motivated by the case of companion forms which Gross hadn't dealt with, and I think he said he had a proof without log schemes just for this case, and then Felipe came along, added the log schemes, and re-did the general case :-) $\endgroup$
    – Kevin Buzzard
    Jan 23, 2010 at 9:55
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Danny Gillam at Brown also has some nice notes on his webpage.

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