ag.algebraic geometry - Reference Request for Drinfeld and Laumon Compactifications

Background

Let $X$ denote a smooth projective curve over $mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $mathbb{C},$ which has associated flag variety $G/B.$

Then we can consider the variety $Maps^d(X, G/B)$ of maps from $X$ to $G/B$ of fixed degree $d$ where $d$ is an $mathbb{N}$-linear combination of coroots of $G.$ See the top of page 2 of this paper by Alexander Kuznetsov Kuznetsov for the definition of degree. The Plucker embedding of the flag variety into projective space gives an alternative formulation of $Maps^d(X, G/B)$ which can be found in section 1.2 of Kuznetsov or in this survey article of Alexander Braverman Braverman.

In general, $Maps^d(X, G/B)$ is not compact, but there is a compactification due to Drinfeld, which is referred to as the variety of quasi-maps and denoted $QMaps^d(X, G/B).$ See Kuznetsov or Braverman.

On the other hand, when $G = SL_n,$ there is a second compactfication due to Laumon. This is because when $G = SL_n,$ we have both the Plucker embedding description of the flag variety, but also the description of the flag variety as flags of vector spaces. This latter description gives another formulation of $Maps^d(X, G/B)$ but leads to a compactification known as quasi-flags. Once again, see Kuznetsov. When $n>2,$ varieties of quasi-maps and of quasi-flags are different. It turns out that quasi-flags are always smooth, while quasi-maps have singularities.

Broadening our focus somewhat, we could instead consider the representable map of stacks $Bun_B(X) to Bun_G(X),$ and note that the fiber over the trivial $G$-bundle is the union of all the $Maps^d(X, G/B)$ for all possible degrees (note that the degree just tells us which connected component of $Bun_B$ we live in).

Just as the variety of maps above was not compact, the map $Bun_B to Bun_G$ is not proper. But there exists a relative compactification of $Bun_B,$ also referred to as the Drinfeld compactification, which I will denote $Bun_B^D.$ This compactification still maps to $Bun_G,$ but the map is now proper. The fiber over the trivial bundle of this map coincides with the union of all $QMaps^d(X, G/B).$

As before, when $G = SL_n,$ there is a second compactification of $Bun_B$ which I will denote $Bun_B^L$ whose fiber over the trivial bundle coincides with the union of all the quasi-flags varieties. See this paper by Braverman and Gaitsgory BG or this follow-up paper by Braverman, Gaitsgorgy, Finkelberg, and Mirkovic BGFK for more details.

Question

In Kuznetsov, Kuznetsov proves that when $X = mathbb{P}^1$ and $G = SL_n,$ there is a map from the space of quasi-flags of degree $d$ to the space of quasi-maps of degree $d$ which is a small resolution of singularities.

Later, in BG, it is asserted that Kuznetsov proved that $Bun_B^L(X)$ is a small resolution of singularities of $Bun_B^D(X)$ for any smooth projective curve $X.$

It seems to me that there are two discrepancies here. One has to do with an arbitrary smooth projective curve versus $mathbb{P}^1.$ The second has to do with moving from the varieties of quasi-maps and quasi-flags to the stacks $Bun_B^D$ and $Bun_B^L.$

Does anyone know a reference which explains the bridge between Kuznetsov and the assertions of BG? Or perhaps this was just something clear to the experts which never warranted an explanation?

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