Cartan theorems A and B

E484523

mathematical theorem result in complex analytic geometry

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.

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All labels observed (6)

Label	Occurrences
Cartan theorems A and B canonical	3
Cartan–Serre theory of coherent analytic sheaves	2
"Coherent Analytic Sheaves"	1
Cartan theorem A	1
Cartan theorem B	1
Cartan’s theorems A and B	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]

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
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 ⓘ linked to: Cartan theorems A and B Cartan theorem B ⓘ linked to: Cartan theorems A and B
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 ⓘ Serre’s GAGA theorem ⓘ linked to: Serre’s cohomological methods in algebraic geometry
involvesConcept	coherent sheaf ⓘ global section ⓘ holomorphic function ⓘ sheaf cohomology ⓘ
mathematicalDomain	algebraic topology ⓘ analysis ⓘ geometry ⓘ
namedAfter	Henri Cartan ⓘ
provenBy	Henri Cartan ⓘ
relatedTo	Grauert’s theorem ⓘ linked to: Oka–Weil theorem Oka–Cartan theory ⓘ Stein manifold ⓘ coherent algebraic sheaf ⓘ
usedFor	Oka–Cartan theory ⓘ 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