Lefschetz hyperplane theorem

E420792

mathematical theorem theorem in algebraic geometry theorem in topology

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.

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All labels observed (8)

Label	Occurrences
Lefschetz hyperplane theorem canonical	4
Lefschetz (1,1)-theorem	1
Lefschetz hyperplane theorem for homology	1
Lefschetz hyperplane theorem for homotopy	1
Lefschetz theorems	1
relative Lefschetz hyperplane theorem	1
weak Lefschetz hyperplane theorem	1
weak Lefschetz 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]

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
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 ⓘ relative Lefschetz hyperplane theorem ⓘ linked to: Lefschetz hyperplane theorem
hasVersion	Lefschetz hyperplane theorem for homology ⓘ linked to: Lefschetz hyperplane theorem Lefschetz hyperplane theorem for homotopy ⓘ linked to: Lefschetz hyperplane theorem strong Lefschetz hyperplane theorem ⓘ linked to: Hard Lefschetz theorem weak Lefschetz hyperplane theorem ⓘ linked to: Lefschetz hyperplane theorem
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 ⓘ
provedBy	Solomon Lefschetz ⓘ
relatedTo	Hard Lefschetz theorem ⓘ Lefschetz decomposition ⓘ Lefschetz fixed-point theorem ⓘ Lefschetz pencil ⓘ
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