All Questions
Tagged with geometric-langlands ag.algebraic-geometry
37 questions
40
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1
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Implications and consequences of the recent proof of the geometric Langlands conjecture
I am a beginner in mathematical physics and geometric Langlands, having very limited knowledge in both fields so far.
The proof of geometric Langlands conjecture is published a few months ago. What ...
- 203
9
votes
0
answers
963
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Geometrization of the global Langlands correspondence?
Fargues-Scholze famously describe arithmetic local Langlands via global geometric Langlands on the Fargues-Fontaine (FF) curve.
The FF curve acts like an algebraic curve over $\mathbb{C}_p$ (its ...
- 15.4k
8
votes
0
answers
644
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Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
- 69
3
votes
0
answers
128
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Smooth unipotent algebraic groups over $\mathbb A^n$
Let $G\to \mathbb A^n_{\mathbb C}$ be a smooth morphism whose fibers at any point of $\mathbb A^n$ are unipotent groups. Can we conclude that $G\simeq \mathbb A^{n+N}_{\mathbb C}$ for some $N$, as a ...
- 455
3
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0
answers
202
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What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 24.8k
12
votes
1
answer
1k
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Relation between motives and geometric Langlands
When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have ...
7
votes
1
answer
419
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Kapustin-Witten branes and the derived moduli stack of Higgs bundles
A lot has been discussed on overflow regarding geometric Langlands and the physics of Kapustin and Witten's groundbreaking paper https://arxiv.org/abs/hep-th/0604151. I would like to add my two cents ...
- 163
1
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0
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162
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Definition of nearby cycle over an affine line
In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...
- 291
4
votes
0
answers
325
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In which sense affine Grassmannian is "affine"
A pretty naïve question: Which meaning has the term "affine" in the notion of affine Grassmanian. Especially, I do not see any immediate connection to the concept of an "affine scheme&...
- 5,998
4
votes
0
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270
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Confusion about definition of crystals
In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...
- 426
2
votes
1
answer
164
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What is the sum operation on torsors induced by Weil uniformization?
Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
- 1,969
12
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0
answers
2k
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Roadmap to geometric Langlands for a mathematical physics student
I am a student of both mathematics and physics, who has recently been studying string theory. My mathematics background is mostly differential geometry (principal bundles, Lie groups, etc.), although ...
- 349
2
votes
1
answer
268
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Duality of Hitchin fibrations in type A
For $G = GL_n$, it is known that the generic fibers of the Hitchin fibration are the Picard stacks of line bundles on the corresponding spectral curves and the duality of Hitchin fibrations in this ...
- 2,216
2
votes
1
answer
183
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Generation of trace fields of Frobenii on local systems
Let $\overline{X}$ be a smooth proper curve over $\mathbb{F}_q$, for some $q$, $S$ a collection of $\mathbb{F}_q$ points of $\overline{X}$, and set $X=\overline{X}-S$.
For a rank $n$ $\overline{\...
- 254
15
votes
1
answer
846
views
Beilinson-Drinfeld local geometric class field theory
There is the following version of categorical local geometric class field theory:
Let $\mathbb{D}=\operatorname{Spec} \mathbb{C}((t))$, $L\mathbb{G}_m$: the loop group of $\mathbb{G}_m$ over $\mathbb{...
- 647
2
votes
0
answers
155
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Integral kernels for geometric langlands
My apologies for the imprecise question(s), it should be clear enough that I´m a complete beginner in this subject.
The (de Rham) Geometric Langlands Conjecture over $\mathbb{C}$ takes as input a ...
user108998
4
votes
0
answers
229
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Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?
In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...
- 2,044
8
votes
0
answers
1k
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Ramified Geometric Langlands
Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...
- 2,751
7
votes
1
answer
538
views
Remark 12.8.8 in Arinkin--Gaitsgory
I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...
- 723
7
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1
answer
555
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Examples of function fields Langlands for small genus (<= 2)
See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...
- 2,216
20
votes
1
answer
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LMS Lectures on Geometric Langlands
Everybody knows how insightful are David Ben-Zvi talks (and comments/answers here on mathoverflow). I was trying to watch the LMS 2007 Lecture Series on Geometric Langlands by David, supposedly made ...
- 653
24
votes
1
answer
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What do Hecke eigensheaves actually look like?
Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
- 148k
7
votes
1
answer
872
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What are local spaces and what are they good for?
Factorization structures have been popular in the past decade. Recently a variant of this structure has been suggested by Ivan Mirkovic (and possibly collaborators). This variant, which goes under the ...
- 2,751
11
votes
2
answers
1k
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Examples of Eigensheaves outside of langlands
In geometric Langlands, one looks at correspondences of the form
$$ Bun_n(X) \leftarrow Hecke \rightarrow X\times Bun_n(X)$$
and calls a sheaf on the lefthand space Hecke eigensheaf, if pulling ...
- 13.2k
3
votes
0
answers
334
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Local counterpart of the NON-Hitchin Hecke eigen-sheaves ?
Insight of Beilinson and Drinfeld at early 90-ies - that Hitchin's D-modules are Hecke eigen-D-modules. However they are NOT all Hecke-eigensmodules and actually they are only the half-dimensional ...
- 24.8k
15
votes
1
answer
2k
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What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ?
What is the relation between Lafforgue's result on Langlands
and Frenkel-Gaitsgory-Vilonen ? ( http://arxiv.org/abs/math/0012255 , http://arxiv.org/abs/math/0204081 )
Does one imply other ? If not ...
- 24.8k
2
votes
0
answers
408
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Orbit stratification of semi infinite flag manifold?
Denote semi infinite flag manifold by $Fl_{\infty/2}=G((t))/N_-((t))H[[t]]$, denote $B_-((t))=N_-((t))H[[t]]$
from the book of Frenkel and Benzvi" Vertex algebras and algebraic curves", They take ...
- 5,505
4
votes
2
answers
1k
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Opers, connections
My questions here are from my attempt at trying to understand the definition on pg 15 in [FG2]-"Local Geometric Langlands Correspondence & Affine Kac-Moody Algebras" (http://arxiv.org/PS_cache/...
- 2,216
11
votes
3
answers
1k
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The affine Grassmannian and the Bogomolny equations
In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more ...
- 44.7k
6
votes
0
answers
498
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Generalizations of Drinfeld Symmetric Space? (Drinfeld homogeneous space, Drinfeld flag variety?)
Are there natural generalizations of the Drinfeld symmetric space? For $\mathbb{K}$, a non-Archimedean local field, the Drinfeld symmetric space can be defined as the complement of all $\mathbb{K}$-...
- 281
21
votes
3
answers
2k
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A good example of a curve for geometric Langlands
I'm currently working through Frenkel's beautiful paper:
http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf.
I'm looking for a good example of a projective curve to get my hands dirty, and go ...
- 2,216
2
votes
1
answer
658
views
Understanding formula in Frenkel-Witten
I'm not the person to understand everything in Geometric Endoscopy and Mirror Symmetry, but some parts of it are reasonably clear to me.
In particular, one of the main objects, mathematically ...
- 15.1k
50
votes
1
answer
15k
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Consequences of Geometric Langlands
So, lots of people work on the Geometric Langlands Conjecture, and there have been a few questions around here on it (admittedly, several of them mine). So here's another one, tagged community wiki ...
4
votes
1
answer
320
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Reverse Langlands transform
What os the meaning of a reverse Langlands transform to which Drinfeld seems to refer?
- 15.1k
16
votes
3
answers
5k
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What is an Oper?
Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems. I keep reading that there's a substack of "opers" but am having trouble locating a definition....
- 16k
12
votes
3
answers
3k
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ubiquitous quantum cohomology
Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...
- 10.8k
18
votes
7
answers
6k
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Langlands Dual Groups
Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does ...
- 16k