Serre vanishing theorem
E883473
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Serre vanishing theorem canonical | 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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Kodaira vanishing theorem NERFINISHED ⓘ Serre duality NERFINISHED ⓘ Serre’s cohomological criterion for ampleness NERFINISHED ⓘ Serre’s theorem on projective schemes and graded rings NERFINISHED ⓘ |
| 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)
NERFINISHED
ⓘ
Hartshorne’s Algebraic Geometry NERFINISHED ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Serre vanishing theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.