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Compact generation of the category of D-modules on the stack of \(G\)-bundles on a curve. (English) Zbl 1342.14041

Let \(k\) be a ground field of characteristic zero, \(X\) be a smooth proper curve, and \(G\) be a connected reductive group. The authors prove that the differential-graded category \(C=\text{D-mod}(\mathcal{Y})\) of D-modules on the Artin stack \(\mathcal{Y}=\text{Bun}_G\) of principal \(G\)-bundles \(P\rightarrow X\) is compactly generated. In other words, there is a generating system of compact objects \(c_\alpha\in C\), which means that the Hom functors \(\text{Hom}(c_\alpha,-)\) commutes with arbitrary sums, and that \(\text{Hom}(c_\alpha,c)=0\) for all \(c_\alpha\) only if \(c\) is the zero-object.
This property is easily seen for quotient stacks \(\mathcal{Y}=[Z/H]\), where \(Z\) is a quasicompact scheme and \(H\) is an algebraic group. More generally, it is valid for quasicompact Artin stacks \(\mathcal{Y}\) whose field-valued points have affine automorphism groups, according to []. However, the property usually fails if \(\mathcal{Y}\) is not quasicompact, even for schemes. An example given in this paper is the smooth locally algebraic surface \(X=\bigcup U_i\) with \(U_i=X_i\smallsetminus\left\{x_i\right\}\), where \(\ldots\rightarrow X_2\rightarrow X_1 \rightarrow X_0\) is a suitable sequence of blowing ups with centers \(x_i\in X_i\): Here the DG-category of D-modules is not compactly generated. So for \(\mathcal{Y}=\text{Bun}_G\), the result is rather surprising.
The authors introduce an abstract sufficient criterion, called truncatability, for \(\text{D-Mod}(\mathcal{Y})\) to be compactly generated. The relevant terminology is as follows: A closed substack \(i:\mathcal{Z}\subset\mathcal{Y}\) is called truncative if the functor \(i^! \) sends compact objects in the category of D-modules on \(\mathcal{Y}\) to compact objects in the category of D-modules on \(\mathcal{Z}\). There are several other characterizations, in terms of related functors \(i_{\text{dR},*}\), as well as \(j^*\) and \(j_*\), where \(j:U\rightarrow\mathcal{Y}\) is the complementary open substack. One equivalent condition is that \(j_*\) admits a continuous right adjoint. In this situation, one also says that \(U\) is co-truncative. The concept also extends to locally closed substacks. An Artin stack \(\mathcal{Y}\) is called QCA if it is quasicompact, and its field-valued points have affine automorphism groups. Without the former condition, on says that \(\mathcal{Y}\) is locally QCA. In this situation, the authors call \(\mathcal{Y}\) truncatable if it is covered by open quasicompact substacks that are co-truncative in the above sense, and then show that the ensuing category of D-modules is compactly generated.
In the case \(G=\text{SL}_2\), it is then established that the open substacks \(\text{Bun}_G^{(n)}\) of vector bundles that do not admit line subbundles of degree \(>n\) form open co-truncative substacks that cover \(\mathcal{Y}=\text{Bun}_G\). This is generalized to arbitrary connected reductive \(G\), by considering the rational cone \(\Lambda_G^{+,\mathbb{Q}}\) generated by the dominant coweights, with respect to a fixed choice of Borel subgroup. For each \(\theta\) in this cone, the authors consider the open substack \(\text{Bun}_G^{\leq \theta}\) of all bundles \(\mathcal{P}_G\) so that for each reduction \(\mathcal{P}_P\) so some standard parabolic \(P\subset G\) of degree \(\mu\) (defined in a suitable way), one has \(\mu\leq \theta\) (for some natural order relation coming from the positive coroots). The authors are then able to show that this gives the desired co-trucative open substacks, provided that \(\theta\) is large enough, measured in terms of the simple roots of the reductive group \(G\) and the genus of the smooth curve \(X\).

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14D23 Stacks and moduli problems
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