Hirzebruch–Riemann–Roch theorem

E259772

mathematical theorem theorem in algebraic geometry theorem in topology

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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All labels observed (2)

Label	Occurrences
Hirzebruch–Riemann–Roch theorem canonical	8
Hirzebruch–Riemann–Roch for smooth projective varieties	1

How this entity was disambiguated

This entity first appeared as the object of triple T2364564 — 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: Hirzebruch–Riemann–Roch theorem
Context triple: [Riemann–Roch theorem, generalizedBy, Hirzebruch–Riemann–Roch 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. 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.
C. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
D. Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
E. 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.
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: Hirzebruch–Riemann–Roch theorem
Target entity description: 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.

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. 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.
C. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
D. Atiyah–Bott fixed-point theorem
The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
E. 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.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in algebraic geometry ⓘ theorem in topology ⓘ
appliesTo	compact complex manifolds ⓘ smooth projective varieties ⓘ
assumption	compactness of the manifold ⓘ complex structure on the manifold ⓘ smoothness of the variety ⓘ
category	Atiyah–Singer index theorem ⓘ surface form: index theorems
domainObject	compact complex manifold X ⓘ holomorphic vector bundle E over X ⓘ
expresses	holomorphic Euler characteristic as integral of characteristic classes ⓘ
field	K-theory ⓘ algebraic geometry ⓘ complex geometry ⓘ differential topology ⓘ
generalizes	Riemann–Roch theorem ⓘ Riemann–Roch theorem ⓘ surface form: Riemann–Roch theorem for curves Riemann–Roch theorem ⓘ surface form: Riemann–Roch theorem for divisors on algebraic curves
givesFormulaFor	holomorphic Euler characteristic of a coherent sheaf ⓘ holomorphic Euler characteristic of a vector bundle ⓘ
hasConsequence	relations between Chern numbers and Euler characteristics ⓘ topological formulas for arithmetic genera ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	topological invariance of holomorphic Euler characteristic ⓘ
inspired	Grothendieck–Riemann–Roch theorem ⓘ
namedAfter	Friedrich Hirzebruch ⓘ
provedBy	Friedrich Hirzebruch ⓘ
publishedIn	Mathematische Annalen ⓘ
relatedTo	Atiyah–Singer index theorem ⓘ Grothendieck–Riemann–Roch theorem ⓘ Noether’s formula ⓘ Hirzebruch signature theorem ⓘ surface form: signature theorem
relatesConcept	Chern classes ⓘ Todd class ⓘ characteristic classes ⓘ cohomology ⓘ complex vector bundles ⓘ holomorphic Euler characteristic ⓘ topological K-theory ⓘ
statementForm	χ(X,E) = ∫_X ch(E)·Td(TX) ⓘ
usedIn	classification of complex surfaces ⓘ computation of dimensions of spaces of sections ⓘ enumerative geometry ⓘ study of moduli spaces ⓘ
uses	Chern character of a vector bundle ⓘ Todd class of the tangent bundle ⓘ
yearProved	1954 ⓘ

How these facts were elicited

Referenced by (9)

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

Riemann–Roch theorem → generalizedBy → Hirzebruch–Riemann–Roch theorem ⓘ

Atiyah–Singer index theorem → generalizes → Hirzebruch–Riemann–Roch theorem ⓘ

Friedrich Hirzebruch → knownFor → Hirzebruch–Riemann–Roch theorem ⓘ

Chern classes → usedFor → Hirzebruch–Riemann–Roch theorem ⓘ

Grothendieck–Riemann–Roch theorem → generalizes → Hirzebruch–Riemann–Roch theorem ⓘ

Grothendieck–Riemann–Roch theorem → hasConsequence → Hirzebruch–Riemann–Roch theorem ⓘ

this entity surface form: Hirzebruch–Riemann–Roch for smooth projective varieties

Grothendieck–Riemann–Roch theorem → relatedTo → Hirzebruch–Riemann–Roch theorem ⓘ

Chern character → usedIn → Hirzebruch–Riemann–Roch theorem ⓘ

Todd class → roleIn → Hirzebruch–Riemann–Roch theorem ⓘ

All labels observed (2)

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Statements (48)

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Referenced by (9)

How this entity was disambiguated

How these facts were elicited