Serre vanishing theorem

E883473

theorem in algebraic geometry

The Serre vanishing theorem is a fundamental result in algebraic geometry stating that, on a projective variety, sufficiently high tensor powers of an ample line bundle have vanishing higher cohomology groups.

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All labels observed (2)

Label	Occurrences
Serre vanishing theorem canonical	2
Serre vanishing	1

How this entity was disambiguated

This entity first appeared as the object of triple T10732820 — 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: Serre vanishing theorem
Context triple: [Serre duality, relatedConcept, Serre vanishing theorem]

A. Lefschetz hyperplane theorem
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.
B. Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
C. 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.
D. 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.
E. Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
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: Serre vanishing theorem
Target entity description: The Serre vanishing theorem is a fundamental result in algebraic geometry stating that, on a projective variety, sufficiently high tensor powers of an ample line bundle have vanishing higher cohomology groups.

A. Lefschetz hyperplane theorem
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.
B. Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
C. 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.
D. 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.
E. Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	theorem in algebraic geometry ⓘ
appliesTo	coherent sheaf ⓘ projective variety ⓘ
asserts	vanishing of higher cohomology for large tensor powers of an ample line bundle ⓘ
assumes	ample invertible sheaf ⓘ coherent sheaf on a projective scheme ⓘ
category	cohomological vanishing theorem ⓘ
concerns	asymptotic behavior of cohomology under twisting by ample line bundles ⓘ
concludes	H^i(X, F \otimes L^{\otimes n}) = 0 for all i > 0 and n \gg 0 ⓘ higher cohomology groups eventually vanish for large twists by an ample line bundle ⓘ
context	projective scheme over a Noetherian ring ⓘ projective variety over a field ⓘ
field	algebraic geometry ⓘ
generalizes	vanishing of higher cohomology for sufficiently positive divisors on curves ⓘ
holdsFor	ample line bundle on a projective scheme ⓘ twists of coherent sheaves by high powers of an ample line bundle ⓘ
holdsOver	Noetherian base ring ⓘ
implies	eventual surjectivity of restriction maps of global sections in some settings ⓘ global generation of sufficiently high tensor powers of an ample line bundle under additional hypotheses ⓘ
involves	ample line bundle ⓘ higher cohomology group ⓘ sheaf cohomology ⓘ
namedAfter	Jean-Pierre Serre ⓘ
prerequisiteFor	cohomology and base change results on projective schemes ⓘ construction of Proj of a graded ring as a projective scheme ⓘ
quantifier	there exists n_0 such that for all n \ge n_0 higher cohomology vanishes ⓘ
relatedTo	Castelnuovo–Mumford regularity ⓘ Kodaira vanishing theorem ⓘ Serre duality ⓘ Serre’s cohomological criterion for ampleness ⓘ linked to: Serre’s theorem on projective embeddings via ample line bundles Serre’s theorem on projective schemes and graded rings ⓘ linked to: Serre’s theorem on projective embeddings via ample line bundles
status	standard foundational result in modern algebraic geometry ⓘ
strengthens	basic finiteness theorems for cohomology on projective schemes ⓘ
toolIn	birational geometry ⓘ minimal model program ⓘ study of positivity of line bundles ⓘ
typicalFormulation	If X is projective over a Noetherian ring and L is ample, then for any coherent sheaf F on X there exists n_0 such that H^i(X, F \otimes L^{\otimes n}) = 0 for all i > 0 and n \ge n_0 ⓘ
usedFor	Castelnuovo–Mumford regularity theory ⓘ cohomological dimension estimates ⓘ construction of projective embeddings via very ample line bundles ⓘ embedding projective schemes into projective space ⓘ finiteness of graded modules of sections ⓘ proving Serre’s theorem on projective normality ⓘ regularity results in algebraic geometry ⓘ
usedIn	EGA (Éléments de géométrie algébrique) ⓘ linked to: Éléments de géométrie algébrique Hartshorne’s Algebraic Geometry ⓘ linked to: Hartshorne Algebraic Geometry

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Serre duality → relatedConcept → Serre vanishing theorem ⓘ

Serre’s theorem on projective embeddings via ample line bundles → involvesConcept → Serre vanishing ⓘ

linked to: Serre vanishing theorem

Serre’s theorem on projective embeddings via ample line bundles → implies → Serre vanishing theorem ⓘ

All labels observed (2)

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Statements (46)

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Referenced by (3)

How this entity was disambiguated

How these facts were elicited