Koszulness of some DG-algebras and a paper by Kohno and Oda

Question

This is a follow-up of my previous question Formality of the 2nd ordered configuration space of a closed Riemann surface.

At page 131 of [B], R. Bezrukavnikov states Proposition 4.1, in which he shows (among other things) that a certain enveloping algebra $U(L)$, related to the topology of the $n$th configuration space $X_n$ (where $X$ is a smooth compact Riemann surface of genus $g \geq 1$) is a Koszul quadratic algebra if and only if $n \leq 2$.

Then he makes this cryptic remark (see p. 133):

An equivalent form of the incorrected statement "the graded algebra $U(L)$ is a Koszul quadratic algebra" was stated $($and "proved"$)$ in [KO],

saying nothing more on the subject.

I would like to use these results in my research, but I am not a specialist in the field. So let me ask the following

Q. What is the "equivalent statement" in [KO] Bezrukavnikov refers to, and what are (if any) the mistakes in its proof?

Bibliography.

[B] R. Bezrukavnikov: Koszul DG-algebras arising from configuration spaces, Geometric and functional analysis 4 (1994), 119-135.

[KO] T. Kohno, T. Oda: The lower central series of the pure braid group of an algebraic curve, Advances in Pure Math 12 (1987), 201-219.

Answer 1

(Just for notation, I let $\mathrm{Conf}_n(X) = \{ x \in X^n \mid \forall i \neq j,\, x_i \neq x_j \}$. This is what Bezrukavnikov calls $X_n$ and what Kohno–Oda call $F_{0,n}(X)$.)

The issue is in §3, on page 208, when they claim that the monodromy representation $\rho : \pi_1(\mathrm{Conf}_n(\Sigma_g) \to \mathrm{Aut}(\pi_1(\Sigma_g - \{p_1, ..., p_n\})^{ab})$ is trivial. They claim that $\pi_1(\mathrm{Conf}_{n+1}(\Sigma_g)) \to \pi_1(\mathrm{Conf}_n(\Sigma_g))$ has a section. Unfortunately, this is not correct if $g \ge 2$ and $n \ge 2$. (For the torus it exists because it's a Lie group so you can define one explicitly, see "Configuration spaces" by Fadell–Neuwirth.)

Here I should give some credit as I didn't find the issue on my own: see the last remark in "Braids on surfaces and finite type invariants" by Bellingeri and Funar, and a remark in Hain's "Infinitesimal presentations of the Torelli groups".

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To see which statement is equivalent to the Koszulity of the universal enveloping algebra, I'll try to rephrase everything in slightly more "modern" language (at the very least, language that I am more familiar with). Bezrukavnikov uses a convention (the same as in the book Quadratic Algebras of Polishchuk–Positselski) that a dg-algebra $A$ is quadratic if it is generated by $A^1$ and the kernel of $T(A^1) \to A$ is generated by elements in degree $2$. If $A$ is graded commutative, then the Koszul dual $A^!$ is the enveloping algebra of $\mathcal{L}$, where $\mathcal{L}$ is the free Lie algebra on the dual $(A^1)^\vee$ modded out by the image of the dual of the product $(A^2)^\vee \to (A^1)^\vee \otimes (A^1)^\vee$. In the case where $A = H^*(X)$, this is exactly the holonomy Lie algebra $\mathfrak{g}_X$ defined by Kohno–Oda.

There are various equivalent definitions of Koszulness of a quadratic DGA $A$. One of them is that the Koszul complex $(A \otimes A^¡, d)$ is acyclic, where $A^¡$ is dual to $A^!$. The Koszul complex of $U(\mathfrak{g}_X)$ is exactly the one called $R(X)$, and Kohno–Oda prove that it is acyclic in the proof of Lemma 4.1, using the section claimed above.

(Maybe a quick remark: nowadays, algebras which aren't $1$-generated can also be considered Koszul. Instead of asking $A$ to be generated by $A^1$ with relations in degree $2$, you can ask $A$ to be generated by some set of generators and ask for relations of weight $2$ with respect to the word length. Then whenever you see an $\operatorname{Ext}$ or something, it is weight graded, and whenever there is a condition of the type "generated in degree $1$" you can replace it by "generated in weight $1$".)