Hitchin fibration
E886932
The Hitchin fibration is a fundamental geometric structure in the theory of Higgs bundles that organizes their moduli space into an algebraically completely integrable system with deep connections to representation theory and the geometric Langlands program.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Hitchin fibration canonical | 3 |
| Hitchin fibration for Langlands dual group | 1 |
| Hitchin map | 1 |
| Hitchin moduli space | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829550 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hitchin fibration Context triple: [Nigel Hitchin, notableFor, Hitchin fibration]
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A.
Beilinson–Drinfeld Grassmannian
The Beilinson–Drinfeld Grassmannian is a geometric object in algebraic geometry and representation theory that generalizes the affine Grassmannian to configurations of multiple points, playing a central role in the geometric Langlands program.
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B.
Beilinson–Bernstein localization theorem
The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.
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C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
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D.
Hodge filtration
The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
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E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hitchin fibration Target entity description: The Hitchin fibration is a fundamental geometric structure in the theory of Higgs bundles that organizes their moduli space into an algebraically completely integrable system with deep connections to representation theory and the geometric Langlands program.
-
A.
Beilinson–Drinfeld Grassmannian
The Beilinson–Drinfeld Grassmannian is a geometric object in algebraic geometry and representation theory that generalizes the affine Grassmannian to configurations of multiple points, playing a central role in the geometric Langlands program.
-
B.
Beilinson–Bernstein localization theorem
The Beilinson–Bernstein localization theorem is a fundamental result in geometric representation theory that realizes representations of semisimple Lie algebras as sheaves of differential operators on flag varieties, establishing an equivalence between algebraic and geometric categories.
-
C.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
D.
Hodge filtration
The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraically completely integrable system
ⓘ
fibration ⓘ geometric structure ⓘ |
| actsOn | moduli space of Higgs bundles ⓘ |
| baseDimension | half the dimension of the Higgs moduli space ⓘ |
| codomain | Hitchin base identified with space of invariant polynomials valued differentials ⓘ |
| constructedFrom |
characteristic polynomial of the Higgs field
ⓘ
invariant polynomials of a complex reductive Lie algebra ⓘ |
| context |
complex reductive algebraic group G
ⓘ
smooth projective algebraic curve over complex numbers ⓘ |
| definedOn |
moduli space of semistable Higgs bundles
ⓘ
moduli space of stable Higgs bundles ⓘ |
| describedIn | Stable bundles and integrable systems (Hitchin, 1987) ⓘ |
| domain | moduli space of G-Higgs bundles on a smooth projective curve ⓘ |
| dualTo |
Hitchin fibration for Langlands dual group
ⓘ
linked to:
Hitchin fibration
|
| field |
algebraic geometry
ⓘ
differential geometry ⓘ mathematical physics ⓘ representation theory ⓘ |
| genericFiberDimension | half the dimension of the Higgs moduli space ⓘ |
| hasBase | Hitchin base ⓘ |
| hasComponent | discriminant locus where spectral curve is singular ⓘ |
| hasFiber |
generic fiber is a Jacobian of a spectral curve
ⓘ
generic fiber is a Prym variety in the non-simply connected case ⓘ generic fiber is an abelian variety ⓘ |
| hasGeneralization |
parabolic Hitchin fibration
ⓘ
wild Hitchin fibration ⓘ |
| hasProperty |
algebraically completely integrable
ⓘ
base is an affine space ⓘ completely integrable Hamiltonian system ⓘ defines an integrable system on the moduli of Higgs bundles ⓘ fibers are Lagrangian with respect to natural symplectic form ⓘ generic fibers are torsors under abelian varieties ⓘ |
| hasSymmetry | action of the Picard stack of the spectral curve ⓘ |
| inspired | subsequent work on integrable systems from moduli spaces ⓘ |
| introducedBy | Nigel Hitchin ⓘ |
| introducedIn | 1987 ⓘ |
| mapType |
Lagrangian fibration
ⓘ
algebraic map ⓘ proper map ⓘ |
| namedAfter | Nigel Hitchin ⓘ |
| relatedTo |
Dolbeault moduli space
ⓘ
Higgs bundle ⓘ S-duality in gauge theory ⓘ geometric Langlands program ⓘ mirror symmetry ⓘ moduli of local systems ⓘ non-abelian Hodge theory ⓘ |
| usedIn |
construction of Hecke eigensheaves
ⓘ
proofs and formulations of geometric Langlands duality ⓘ |
How these facts were elicited
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hitchin fibration Description of subject: The Hitchin fibration is a fundamental geometric structure in the theory of Higgs bundles that organizes their moduli space into an algebraically completely integrable system with deep connections to representation theory and the geometric Langlands program.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hitchin fibration
linked to: Hitchin fibration
subject linked to:
“The self-duality equations on a Riemann surface”
subject linked to:
“The self-duality equations on a Riemann surface”
linked to: Hitchin fibration