Cartan theorems A and B
E484523
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cartan theorems A and B canonical | 1 |
| Cartan–Serre theory of coherent analytic sheaves | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4996545 — 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: Cartan theorems A and B Context triple: [Weierstrass preparation theorem, relatedTo, Cartan theorems A and B]
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A.
Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry on complex manifolds.
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B.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
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C.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
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E.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
- 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: Cartan theorems A and B Target entity description: Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
-
A.
Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry on complex manifolds.
-
B.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
-
C.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
D.
Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
-
E.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
- F. None of above. chosen
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
mathematical theorem ⓘ mathematical theorem ⓘ result in complex analytic geometry ⓘ |
| appliesTo |
Stein space
ⓘ
coherent analytic sheaf ⓘ |
| asserts |
every coherent analytic sheaf on a Stein space is generated by its global sections
ⓘ
higher cohomology groups of a coherent analytic sheaf on a Stein space vanish ⓘ |
| characterizes | coherent analytic sheaves on Stein spaces ⓘ |
| describes |
generation of stalks by global sections
ⓘ
vanishing of higher cohomology groups ⓘ |
| field |
cohomology theory
ⓘ
complex analytic geometry ⓘ several complex variables ⓘ sheaf theory ⓘ |
| hasPart |
Cartan theorem A
NERFINISHED
ⓘ
Cartan theorem B NERFINISHED ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| implies |
H^q(X,F)=0 for q>0 when F is coherent on a Stein space X
ⓘ
existence of enough global sections for coherent analytic sheaves on Stein spaces ⓘ |
| influenced |
Oka–Grauert principle
NERFINISHED
ⓘ
Serre’s GAGA theorem NERFINISHED ⓘ |
| involvesConcept |
coherent sheaf
ⓘ
global section ⓘ holomorphic function ⓘ sheaf cohomology ⓘ |
| mathematicalDomain |
algebraic topology
ⓘ
analysis ⓘ geometry ⓘ |
| namedAfter | Henri Cartan NERFINISHED ⓘ |
| provenBy | Henri Cartan NERFINISHED ⓘ |
| relatedTo |
Grauert’s theorem
NERFINISHED
ⓘ
Oka–Cartan theory NERFINISHED ⓘ Stein manifold NERFINISHED ⓘ coherent algebraic sheaf ⓘ |
| usedFor |
Oka–Cartan theory
NERFINISHED
ⓘ
cohomological characterization of Stein spaces ⓘ embedding theorems for Stein manifolds ⓘ solving extension problems for holomorphic functions ⓘ vanishing theorems in complex geometry ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Cartan theorems A and B Description of subject: Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cartan–Serre theory of coherent analytic sheaves