Notes on Geometric Langlands
This page will contain a collection of .pdf files intended to share with the community my current understanding of
Geometric Langlands
(global and local, classical and quantum).
Not all papers that appear in the list of references
are ready to be posted, but most of them exist in one form or another, and you're
welcome to request a .pdf file
from me. Otherwise, I'll be posting them gradually.
Needless to say, comments and corrections of any kind are extremely welcome!
Note on the status of the assertions:
If a statement is designated as a "Theorem", this means that the proof exists
(but may have not been written down yet).
If a statement is designated as a "Quasi-Theorem", this means that I see a way how
to prove it, but haven't thought through the details.
If you are a graduate student or
a post-doc interested in the area, you are welcome to supply a proof and claim the result.
- Asymptotics of geometric Whittaker coefficients (joint with D. Arinkin, last updated March 16, 2015)
- Compact generation of the category of D-modules on Bun(G) (an "official" paper, joint with V. Drinfeld, last updated May 1, 2015)
- Contractibility of the space of rational maps (an "official paper", last updated Jan. 17, 2012)
- The Atiyah-Bott formula for the cohomology of Bun(G) (an "official paper", last updated April 17, 2015)
- The extended Whittaker category (last updated Dec. 2, 2010)
- Categories over the Ran space (last updated Dec. 2, 2010)
- Localization and the long intertwining operator for representations of affine Kac-Moody algebras (joint with S.~Arkhipov, last updated May 15, 2015)
- Generalities on DG categories (last updated Dec. 30, 2012)
- Filtered colimits of categories, by N.~Rozenblyum (last updated Dec. 30, 2012)
- Functors given by kernels, adjunctions and duality (an "official" paper, last updated Nov. 23, 2015)
- Stacks (last updated Aug. 14, 2011)
- Quasi-coherent sheaves on stacks (last updated Aug. 18, 2011)
- Sheaves of categories and the notion of 1-affineness (an ``official" paper, last updated Aug. 9, 2014)
- Ind-coherent sheaves (an "official" paper, last updated Oct. 14 , 2012)
- DG Indschemes (an "official" paper, joint with N. Rozenblyum, last updated June 23, 2013)
- Crystals and D-modules (an "official" paper, joint with N. Rozenblyum, last updated Oct. 1, 2014)
- On some finiteness questions for algebraic stacks
(an "official" paper, joint with V. Drinfeld, last updated Oct. 26, 2012)
Chapters from the book project 'A study in derived algebraic geometry' by D. Gaitsgory and N. Rozenblyum (prelim version)
- Part I.1: Some higher algebra
(an introduction to oo-categories and review of Lurie's books Nov. 12 , 2015)
- Introduction to Part II (Ind-coherent sheaves) (June 29, 2015)
- Part II.1: Ind-coherent sheaves on schemes
(introduces and studies elementary properties of IndCoh on schemes, June 28 , 2015)
- Part II.2: IndCoh as a functor out of the category of correspondences
(discusses the formalism of IndCoh as a functor of the category of correspondences, Oct. 5, 2015)
- Part II.3: QCoh and IndCoh
(discusses the relationship between QCoh and IndCoh in the framework of correspondences, Aug. 25, 2015)
- Introduction to Part III (Inf-schemes) (June 9, 2015)
- Part III.1: Deformation theory
(sets up deformation theory, the latter being necessary for introducing inf-schemes, Sept. 3, 2015)
- Part III.2: (Ind)-inf-schemes
(defines inf-schemes, which are algebro-geometric
objects that include DG schemes and de Rham prestacks, June 9, 2015)
- Part III.3: Ind-coherent sheaves on (ind)-inf-schemes
(extends the formalism of IndCoh to inf-schemes and discusses its functoriality, Sept. 20, 2015)
- Part III.4: An application: crystals
(explains how the "formalism of 6 operations" for D-modules follows from the theory of IndCoh on inf-schemes, Sept. 20, 2015)
- Introduction to Part IV (Formal geometry) (Sept. 3, 2015)
- Part IV.1: Formal moduli
(reinterprets Lurie's theory of formal moduli problems using the language of inf-schemes, Sept. 20, 2015)
- Part IV.2: Lie algebras and co-commutative co-algebras
(sets up the theory of Le algebras from the point of view of Koszul-Quillen duality, Sept. 3, 2015)
- Part IV.3: Formal groups and Lie algebras
(explains how the pass between Lie algebras and formal groups within the framework of DAG, Sept. 3, 2015)
- Part IV.4: Lie algebroids
(initiates the study of Lie algebroids and modules over them, Sept. 3, 2015)
- Part IV.5: Infinitesimal differential geometry
(studies various aspects of infinitesimal geometry, such as the n-th infinitesimal neighborhood, Sept. 4 , 2015)
- Introduction to Part V (Categories of correspondences) (Sept. 27, 2015)
- Part V.1: The (oo,2)-category of correspondences
(introduces the formalism of correspondences, Oct. 9, 2015)
- Part V.2: Extension theorems for category of correspondences
(extends IndCoh as a functor on the category of correspondences from schemes to inf-schemes, Oct. 5, 2015)
- Part V.3: The (symmetric) monoidal structure on the category of correspondences
(shows how the formalism of correspondences encodes such phenomena as Serre duality, Oct. 2, 2015)
- Introduction to Part A (appendix on (oo,2)-categories) (Oct. 13, 2015)
- Part A.1: Basics of (oo,2)-categories
(defines (oo,2)-categories and introduces some basic constructions, Oct. 9, 2015)
- Part A.2: Straightening and Yoneda (oo,2)-categories
(constructs the straightening/unstraightening procedures and the Yoneda embedding , Oct. 7, 2015)
- Part A.3: Adjunctions in (oo,2)-categories
(studies the procedure of passing to the adjoint 1-morphism, Oct. 9, 2015)