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The minimal model program works with pairs $(X,B)$ where $X$ is a variety and $B$ is a certain kind of divisor on it. I've seen these described as "logarithmic pairs". There are also "log resolutions", "log canonical singularities" and so on.

Question: What do these things have to do with logarithms?

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    $\begingroup$ It would help if you could clarify your background, the length of an answer would seriously depend on it. $\endgroup$
    – Vladimir Dotsenko
    Commented Apr 7, 2022 at 12:19
  • $\begingroup$ I know some algebraic geometry on the level of Hartshorne. I don't know anything about log algebraic geometry, if that's what you're wondering. For what it's worth, I always favor longer and more detailed answers over shorter ones. $\endgroup$
    – Kim
    Commented Apr 7, 2022 at 13:18
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    $\begingroup$ The general idea is that in log geometry goes around the idea of consider differential forms which aren't necessarily regular, bur have certain mild singularities, called logarithmic singularities, along $B$. The reason they are called so is because such forms are related to ones of the form $df/f$, which "morally" are just $d(\log f)$ (except we don't admit $\log f$ itself as a function, just this differential). $\endgroup$
    – Wojowu
    Commented Apr 7, 2022 at 14:07
  • $\begingroup$ Indeed, one of the starting points is the fact that the cohomology of a complement of the normal crossing divisor can be computed via the sheaf of differential forms with logarithmic singularities along the divisor. This is discussed and used in a crucial way in Hodge II, Section 3. $\endgroup$
    – Vladimir Dotsenko
    Commented Apr 7, 2022 at 15:37
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    $\begingroup$ @BenWieland that's a great paper for sure, but logarithmic differential forms do not appear there, instead he uses forms with poles of arbitrary order. As far as I know, it was really Deligne's idea that the logarithmic de Rham complex is quasi-isomorphic to that. $\endgroup$
    – Piotr Achinger
    Commented Apr 7, 2022 at 18:29

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