In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{Li}_s(z)$. How are they related to each other? Any reference would be helpful, but it would be better if it requires less preliminaries.

  • $\begingroup$ Locally, the sheaf of logarithmic forms $\Omega^1(\log D)$ is generated by objects of the form $df/f$, where $f$ vanishes on $D$. In some open covering, there are local representations of this differential form as a logarithmic derivative, and this justifies the name. en.wikipedia.org/wiki/Logarithmic_form $\endgroup$ – Francesco Polizzi Jun 23 '16 at 20:43

I would suggest looking at Hain's article (arXiv version here)

R. Hain. Classical polylogarithms. Motives Proceedings, vol II, Proc. Symposia Pure Math 55.2, 1994.

Very roughly, one writes out a multi-valued function on $\mathbb{P}^1\setminus\{0,1,\infty\}$ which takes values in ${\rm GL}_{n+1}(\mathbb{C})$ where the entries of the matrix are built from (poly)logarithms up to ${\rm Li}_n$. This is the fundamental solution of a linear differential equation. The monodromy of the differential equation gives rise to a local system on $\mathbb{P}^1\setminus\{0,1,\infty\}$. This is the $n$-th polylogarithm local system (which can equivalently be regarded as a sheaf). Actually, doing this for all $n$ gives rise to a pro-local system which you might call the polylogarithm sheaf. Under the correspondence between local systems and representations of the fundamental group, this corresponds to the representation of $\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$ on the completion of its group ring w.r.t. the augmentation ideal.

This has a Hodge-theoretic and even motivic refinement. These are relevant when trying to express regulators on K-theory in terms of polylogarithms (on the way to formulas for special L-values).

You can find all this and more in Hain's article, also lots of references in there. Unfortunately, this is not quite the discussion that would "require less preliminaries"...

  • $\begingroup$ >Under the correspondence between local systems and representations of the fundamental group, this corresponds to... I don't think this is quite right -- I think it's only a rather tiny quotient of the fundamental group. The whole pro-unipotent fundamental group corresponds to so-called "multiple polylogarithms," and the associated differential equation is usually called the "free KZ equation" if I understand these things correctly. $\endgroup$ – Daniel Litt Feb 9 '17 at 4:12

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