# 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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I guess that formally this is not written anywhere, but it is indeed easy to deduce the general case from Kuznetsov's result. The point is that both Drinfeld and Laumon compactifications consist of G-bundles with some kind of degenerate B-structure, where the degeneration occurs at finitely many points of the curve. It is easy to see that the fiber of of the map $Bun_B^L\to Bun_B^D$ depends only on the behaviour of everything in the formal neighbourhood of those points (using this observation you can formally reduce your question to Kuznetsov's result).