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Singular support of coherent sheaves and the geometric Langlands conjecture. (English) Zbl 1423.14085

Given a complex curve \(X\), a reductive group \(G\) and its Langlands dual \(\check{G}\), one can construct two categories which the geometric Langlands programme predicts are equivalent, compatibly with various functors important in representation theory. The first, \[\tag{1} \text{D-mod}(\mathrm{Bun}_G), \] is the derived category of D-modules on the moduli stack of all \(G\)-bundles on \(X\). The second should be something close to the derived category \[ \tag{2} \mathrm{Qcoh}(\mathrm{Loc}_{\check G})\] of quasi-coherent sheaves on the derived stack of all \(\check G\)-local systems on \(X\).
When \(G\) is a torus, geometric Langlands has been proved [G. Laumon, “Transformation de Fourier généralisée”, Preprint, arXiv:alg-geom/9603004; M. Rothstein, Duke Math. J. 84, No. 3, 565–598 (1996; Zbl 0877.14032)]. For other \(G\), however, it has been a long-standing problem to find the correct replacement for the second category (2), since the singularities of \(\mathrm{Loc}_{\check G}\) mean an equivalence between \(\mathrm{Qcoh}(\mathrm{Loc}_{\check G})\) and \(\text{D-mod}(\mathrm{Bun}_G)\) (which is smooth) cannot meet various compatibility conditions.
The paper under review solves this problem by enlarging \(\mathrm{Qcoh}(\mathrm{Loc}_{\check G})\) over the singular locus. To do this, the bulk of the paper develops and studies a notion of “singular support” (in the sense of D. Benson et al. [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 575–621 (2008; Zbl 1171.18007)]) for coherent sheaves on quasi-smooth derived stacks, using the machinery of higher homotopical algebra and \(\infty\)-categories. A sheaf’s singular support lies in an enlargement of the stack given by putting over any singular point the dual of its natural obstruction space. For the quasi-smooth derived stack \(\mathrm{Loc}_{\check G}\) this dual obstruction space is the space of global endomorphisms of the local system, and the authors propose working only with sheaves whose support is over only those endomorphisms which are nilpotent. This is in beautiful geometric analogy with the appearance of nilpotent Arthur parameters in the classical Langlands correspondence.
So the authors’ solution is to take in place of (2) the category of ind-coherent complexes whose singular support lies in the locus of nilpotent endomorphisms. They show this choice of category satisfies numerous properties suggestive of, and consistent with, the geometric Langlands conjecture. In particular they prove compatibility with the geometric Satake equivalence, and show the appropriate Eisenstein functors are defined.
This is a huge, and hugely significant, paper. It is very technical, but it is also beautifully written. In particular, the introduction is really wonderful, serving as a down-to-earth guide to both the paper and how to think about the constructions and results.

MSC:

14D24 Geometric Langlands program (algebro-geometric aspects)
14A20 Generalizations (algebraic spaces, stacks)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
22E57 Geometric Langlands program: representation-theoretic aspects
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References:

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