Noether’s formula
E898499
Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Noether’s formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991982 — 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: Noether’s formula Context triple: [Hirzebruch–Riemann–Roch theorem, relatedTo, Noether’s formula]
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A.
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
Noether’s AF+BG theorem is a foundational result in algebraic geometry that provides conditions under which a polynomial vanishing on the intersection of two plane curves can be expressed as a linear combination of their defining equations.
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B.
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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C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
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D.
Duistermaat–Heckman formula
The Duistermaat–Heckman formula is a result in symplectic geometry that describes how the pushforward of the Liouville measure under a moment map behaves, showing it is piecewise polynomial and linking geometry with equivariant localization techniques.
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E.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
- 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: Noether’s formula Target entity description: Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
-
A.
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
Noether’s AF+BG theorem is a foundational result in algebraic geometry that provides conditions under which a polynomial vanishing on the intersection of two plane curves can be expressed as a linear combination of their defining equations.
-
B.
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.
-
C.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
D.
Duistermaat–Heckman formula
The Duistermaat–Heckman formula is a result in symplectic geometry that describes how the pushforward of the Liouville measure under a moment map behaves, showing it is piecewise polynomial and linking geometry with equivariant localization techniques.
-
E.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
result in complex geometry
ⓘ
theorem in algebraic geometry ⓘ |
| appearsIn |
birational geometry of surfaces
ⓘ
theory of minimal models of surfaces ⓘ |
| appliesTo |
complex algebraic surface
ⓘ
smooth projective surface ⓘ |
| assumes |
surface is defined over the complex numbers
ⓘ
surface is projective ⓘ surface is smooth ⓘ |
| connectedTo |
Chern–Gauss–Bonnet theorem
ⓘ
linked to:
Chern–Weil theory
Enriques–Kodaira classification ⓘ Noether’s inequality ⓘ Todd class ⓘ |
| expresses | holomorphic Euler characteristic in terms of Chern numbers ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ topology of complex surfaces ⓘ |
| hasConsequence |
constraints on possible Chern numbers of surfaces
ⓘ
relations between arithmetic genus and Chern classes ⓘ |
| hasDomain | compact complex surfaces ⓘ |
| hasRole | bridge between analytic and algebraic invariants of surfaces ⓘ |
| historicalPeriod | early 20th century mathematics ⓘ |
| involves |
canonical divisor
ⓘ
first Chern class ⓘ holomorphic Euler characteristic of the structure sheaf ⓘ second Chern class ⓘ structure sheaf ⓘ |
| isSpecialCaseOf | Hirzebruch–Riemann–Roch theorem ⓘ |
| mathematicalArea |
birational classification of surfaces
ⓘ
intersection theory ⓘ sheaf cohomology ⓘ |
| namedAfter | Emmy Noether ⓘ |
| relatedConcept |
Chern numbers c1^2 and c2
ⓘ
arithmetic genus ⓘ geometric genus ⓘ holomorphic Euler characteristic ⓘ irregularity of a surface ⓘ |
| relates |
Chern numbers
ⓘ
holomorphic Euler characteristic ⓘ topological invariants of surfaces ⓘ |
| typeOf |
Riemann–Roch type formula
ⓘ
linked to:
Riemann–Roch theorem
|
| usedFor |
classification of algebraic surfaces
ⓘ
computing invariants of surfaces ⓘ relating geometric and topological data of surfaces ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Noether’s formula Description of subject: Noether’s formula is a fundamental result in algebraic geometry that relates the holomorphic Euler characteristic of a smooth projective surface to its Chern numbers, serving as a special case of the Hirzebruch–Riemann–Roch theorem.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.