Singular support of coherent sheaves and the geometric Langlands conjecture

By D. Arinkin, D. Gaitsgory. 2012

208 citations recorded by the source corpus.

We define the notion of singular support of a coherent sheaf on a quasi-smooth-derived scheme or Artin stack, where “quasi-smooth” means that it is a locally complete intersection in the derived sense. This develops the idea of “cohomological” support of coherent sheaves on a locally complete intersection scheme introduced by D. Benson, S. B. Iyengar, and H. Krause. We study the behavior of singular support under the direct and inverse image functors for coherent sheaves. We use the theory of singular support of coherent sheaves to formulate the categorical geometric Langlands conjecture. We verify that it passes natural consistency tests: It is compatible with the geometric Satake equivalence and with the Eisenstein series functors. The latter compatibility is particularly important, as it fails in the original “naive” form of the conjecture.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ffaa71358c8e60ddaf71a79157502416b745aaa6

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Citations in the corpus

On Some Finiteness Questions for Algebraic Stacks (2011, methodology): “ent Y→IndCoh(Y) is a natural sheaf-theoretic context in its own right. In particular, the category IndCoh(Y) is indispensable to treat the spectral side of the Geometric Langlands correspondence, see [AG]. In this paper the category IndCoh(Y) is used to prove Theorem 0.3.4. More importantly, this ca”
Ind-coherent sheaves (2011, background): “ complete intersection, one can attach a certain subcategory QCoh(X) perf⊂QCoh(X) N ⊂Coh(S) to any Zariski-closed conical subset Nof Spec X(Sym(H1(T ))), where T denotes the tangent complex of X, see [AG] where this theory is developed. 0.1.4. An important feature of the examples considered above is”
On the equivalence between Lurie's model and the dendroidal model for infinity-operads (2013, background): “∆ 1 ⊗(∆1 ⊗C k) is inner anodyne (for every k≥0). For notational simplicity we treat the case k= 2; the higher cases are completely analogous. The tensor product ∆ 1⊗(∆ ⊗C2) is the nerve of the operad [1] ⊗([1] ⊗Ω(C2)). It can be described as the union of all the shuﬄes of the three trees ∆1, ∆1 and ”
Outline of the proof of the geometric Langlands conjecture for $GL_2$ (2013, methodology): “argement of QCoh(LocSysGˇ), whose deﬁnition uses the fact that LocSysGˇ is a derived locally complete intersection, and the theory of singular support of coherent sheaves for such stacks developed in [AG]. 0.2.2. Idea of the proof. The idea of the comparison between the categories D-mod(BunG) and In”