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
NERFINISHED
ⓘ
Enriques–Kodaira classification NERFINISHED ⓘ Noether’s inequality NERFINISHED ⓘ Todd class NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| mathematicalArea |
birational classification of surfaces
ⓘ
intersection theory ⓘ sheaf cohomology ⓘ |
| namedAfter | Emmy Noether NERFINISHED ⓘ |
| relatedConcept |
Chern numbers c1^2 and c2
ⓘ
arithmetic genus ⓘ geometric genus ⓘ holomorphic Euler characteristic ⓘ irregularity of a surface ⓘ |
| relates |
Chern numbers
NERFINISHED
ⓘ
holomorphic Euler characteristic ⓘ topological invariants of surfaces ⓘ |
| typeOf | Riemann–Roch type formula NERFINISHED ⓘ |
| 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
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: 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.