This research paper delves into the intricate realm of algebraic geometry, specifically focusing on the compact generation of DG categories of D-modules. The authors extend a pivotal result by Drinfeld and Gaitsgory, who demonstrated the compact generation of the DG category of D-modules on the moduli stack of principal G-bundles (BunG).
The paper meticulously examines the moduli stack of principal G-bundles with Iwahori level structure (BunI
G), a more complex structure than BunG. The authors establish a stratification of BunI
G analogous to the Harder-Narasimhan stratification of BunG, dissecting it into manageable quasicompact open substacks.
Through a series of technical arguments, the paper demonstrates that the closed complements of these substacks are contractive, implying the co-truncativeness of the open substacks. This property, coupled with the local QCA nature of BunI
G, leads to the central theorem: D-mod(BunI
G) is compactly generated.
Furthermore, the authors assert that their proof seamlessly generalizes to the moduli stack of principal G-bundles with Iwahori level structure at multiple points, Bun(I;x1,...,xk)
G, establishing its compact generation as well.
This result holds significant implications for the broader field of geometric Langlands correspondence. The compact generation of D-mod(BunI
G) serves as a crucial step towards formulating and proving an Iwahori-ramified version of the geometric Langlands conjecture.
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by Taeuk Nam (1... at arxiv.org 11-06-2024
https://arxiv.org/pdf/2411.03057.pdfDeeper Inquiries