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Selecta Mathematica

, Volume 21, Issue 1, pp 1–199 | Cite as

Singular support of coherent sheaves and the geometric Langlands conjecture

  • D. Arinkin
  • D. Gaitsgory
Article

Abstract

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.

Keywords

Cohomological support Geometric Langlands program Derived algebraic geometry Arthur parameters 

Mathematics Subject Classification

14F05 14H60 

Notes

Acknowledgments

The idea of this work emerged as a result of discussions with V. Drinfeld. The very existence of this paper expresses our debt to him. We are grateful to J. Lurie for teaching us the foundations of derived algebraic geometry. We are grateful to N. Rozenblyum for clarifying various aspects of the theory of \({\mathbb {E}}_2\)-algebras. We would like to thank A. Beilinson, H. Krause, V. Lafforgue, A. Neeman, T. Pantev, G. Stevenson for helpful communications and discussions. Special thanks are due to S. Raskin, whose extensive comments on successive drafts of the paper led to substantial improvements. We are very grateful to the anonymous referees for their numerous comments and suggestions. The work of D.A. is partially supported by NSF Grants DMS-1101558 and DMS-0635607. The work of D.G. is partially supported by NSF Grant DMS-1063470.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.CambridgeUSA

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