The Ran-space oper contractibility conjecture

About 13 years old · traced to

Let LocSys⁡\cG\operatorname{LocSys}_\cG be the stack of \cG\cG-local systems and let

sv⁡Ran⁡(X):Op⁡(\cG)Ran⁡(X)glob⁡→LocSys⁡\cG\operatorname{sv}_{\operatorname{Ran}(X)}:\operatorname{Op}(\cG)^{\operatorname{glob}}_{\operatorname{Ran}(X)}\to\operatorname{LocSys}_\cG

be the map from global opers with singularities parametrized by the Ran space. Ran-space oper contractibility conjecture. The fibers of sv⁡Ran⁡(X)\operatorname{sv}_{\operatorname{Ran}(X)} are O\mathcal O-contractible; equivalently,

sv⁡Ran⁡(X)∗:QCoh⁡(LocSys⁡\cG)→QCoh⁡(Op⁡(\cG)Ran⁡(X)glob⁡)\operatorname{sv}_{\operatorname{Ran}(X)}^*:\operatorname{QCoh}(\operatorname{LocSys}_\cG)\to\operatorname{QCoh}(\operatorname{Op}(\cG)^{\operatorname{glob}}_{\operatorname{Ran}(X)})

is fully faithful. The source says this is a strengthening of the oper generation conjecture and a theorem for G=GLnG=GL_n, with evidence for classical groups.

References

Primary source

Dennis Gaitsgory, “Outline of the proof of the geometric Langlands conjecture for GL(2)”, arXiv:1302.2506 (2014).

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