Lefschetz hyperplane theorem
E420792
The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology (especially homology and homotopy groups) of a smooth projective variety to that of its hyperplane sections.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Lefschetz hyperplane theorem canonical | 2 |
| Lefschetz (1,1)-theorem | 1 |
| Lefschetz hyperplane theorem for homology | 1 |
| Lefschetz theorems | 1 |
| relative Lefschetz hyperplane theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4202376 — 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: Lefschetz hyperplane theorem Context triple: [Solomon Lefschetz, knownFor, Lefschetz hyperplane theorem]
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A.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
B.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
C.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
-
D.
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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E.
Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
- 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: Lefschetz hyperplane theorem Target entity description: The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology (especially homology and homotopy groups) of a smooth projective variety to that of its hyperplane sections.
-
A.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
B.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
C.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
-
D.
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.
-
E.
Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in algebraic geometry ⓘ theorem in topology ⓘ |
| appliesTo |
ample divisors
ⓘ
smooth complex projective variety ⓘ |
| assumes |
hyperplane section is smooth
ⓘ
variety is non-singular ⓘ variety is projective ⓘ |
| concerns |
homology groups
ⓘ
homotopy groups ⓘ hyperplane sections ⓘ smooth projective varieties ⓘ |
| context | complex projective space ⓘ |
| field |
algebraic geometry
ⓘ
algebraic topology ⓘ complex geometry ⓘ |
| formalism |
sheaf cohomology
ⓘ
singular homology ⓘ singular homotopy ⓘ |
| generalizedBy |
Grothendieck–Lefschetz theorem
ⓘ
Lefschetz hyperplane theorem self-linksurface differs ⓘ
surface form:
relative Lefschetz hyperplane theorem
|
| hasVersion |
Lefschetz hyperplane theorem
self-linksurface differs
ⓘ
surface form:
Lefschetz hyperplane theorem for homology
Lefschetz hyperplane theorem for homotopy ONNED1 ⓘ strong Lefschetz hyperplane theorem NERFINISHED ⓘ weak Lefschetz hyperplane theorem ONNED1 ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| implies |
isomorphisms of homology groups in low degrees
ⓘ
isomorphisms of homotopy groups in low degrees ⓘ surjectivity of certain homology maps in middle degree ⓘ surjectivity of certain homotopy maps in middle degree ⓘ |
| influenced |
modern algebraic geometry
ⓘ
modern algebraic topology ⓘ |
| namedAfter | Solomon Lefschetz ONNED1 ⓘ |
| provedBy | Solomon Lefschetz NERFINISHED ⓘ |
| relatedTo |
Hard Lefschetz theorem
ⓘ
Lefschetz decomposition ⓘ Lefschetz fixed-point theorem ONNED1 ⓘ Lefschetz pencil ONNED1 ⓘ |
| relates | topology of a projective variety to topology of its hyperplane section ⓘ |
| typicalAssumption | base field is the complex numbers ⓘ |
| usedIn |
Hodge theory
ⓘ
Morse theory on complex varieties ⓘ Picard group computations ⓘ classification of algebraic varieties ⓘ fundamental group calculations ⓘ study of topology of algebraic varieties ⓘ |
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
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: Lefschetz hyperplane theorem Description of subject: The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology (especially homology and homotopy groups) of a smooth projective variety to that of its hyperplane sections.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lefschetz (1,1)-theorem
subject surface form:
SGA 2
this entity surface form:
Lefschetz theorems
subject surface form:
Solomon Lefschetz
this entity surface form:
Lefschetz hyperplane theorem for homology
Lefschetz hyperplane theorem
→
generalizedBy
→
Lefschetz hyperplane theorem
self-linksurface differs
ⓘ
this entity surface form:
relative Lefschetz hyperplane theorem