Riemann–Roch theorem

E47350

mathematical theorem 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.

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All labels observed (8)

Label	Occurrences
Riemann–Roch theorem canonical	10
Riemann–Roch type formulas	2
Riemann–Roch for curves	1
Riemann–Roch formulas	1
Riemann–Roch theorem for curves	1
Riemann–Roch theorem for divisors on algebraic curves	1
Riemann–Roch theorem for families	1
Riemann–Roch theorems	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem ⓘ
appliesTo	compact Riemann surfaces ⓘ smooth projective algebraic curves ⓘ
completedBy	Gustav Roch ⓘ
concerns	divisors on curves ⓘ line bundles on curves ⓘ
expresses	Euler characteristic as degree plus 1 minus genus ⓘ dimension of H^0(L) minus dimension of H^1(L) ⓘ
field	algebraic geometry ⓘ complex analysis ⓘ
generalizedBy	Grothendieck–Riemann–Roch theorem ⓘ Hirzebruch–Riemann–Roch theorem ⓘ Grothendieck–Riemann–Roch theorem ⓘ surface form: Riemann–Roch theorem for higher-dimensional varieties
givesFormulaFor	dimension of the space of global sections of a line bundle ⓘ dimension of the space of meromorphic functions with prescribed zeros and poles ⓘ
hasKeyConcept	Euler characteristic of a line bundle ⓘ Serre duality ⓘ surface form: Serre duality (in modern formulations) canonical divisor ⓘ degree of a divisor ⓘ genus of a curve ⓘ
hasModernFormulationIn	algebraic geometry over arbitrary fields ⓘ sheaf cohomology ⓘ
historicalPeriod	19th century mathematics ⓘ
implies	Riemann–Hurwitz formula in certain cases ⓘ
influenced	development of K-theory ⓘ development of index theorems ⓘ
isFundamentalFor	intersection theory on curves ⓘ theory of divisors on curves ⓘ theory of line bundles on curves ⓘ
namedAfter	Bernhard Riemann ⓘ Gustav Roch ⓘ
originalContext	compact Riemann surfaces ⓘ
originallyProvedBy	Bernhard Riemann ⓘ
relatedTo	Atiyah–Singer index theorem ⓘ
relates	degree of divisors ⓘ dimension of spaces of meromorphic sections ⓘ genus of a curve ⓘ
type	dimension formula ⓘ
usedIn	Brill–Noether theory ⓘ classification of algebraic curves ⓘ coding theory on algebraic curves ⓘ construction of canonical embeddings of curves ⓘ moduli of curves ⓘ study of Jacobian varieties ⓘ study of linear systems on curves ⓘ theory of algebraic function fields ⓘ

Referenced by (18)

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

Bernhard Riemann → knownFor → Riemann–Roch theorem ⓘ

Riemann surfaces → hasTheorem → Riemann–Roch theorem ⓘ

subject surface form: Riemann surface

Georg → notableWork → Riemann–Roch theorem ⓘ

subject surface form: Georg Friedrich Bernhard Riemann

Friedrich → notableConcept → Riemann–Roch theorem ⓘ

subject surface form: Friedrich Bernhard Riemann

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

Chern classes → usedFor → Riemann–Roch theorem ⓘ

this entity surface form: Riemann–Roch theorems

Serre duality → usedFor → Riemann–Roch theorem ⓘ

this entity surface form: Riemann–Roch type formulas

Serre duality → relatedTo → Riemann–Roch theorem ⓘ

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

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

this entity surface form: Riemann–Roch for curves

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

this entity surface form: Riemann–Roch theorem for curves

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

this entity surface form: Riemann–Roch theorem for divisors on algebraic curves

Brill–Noether theory → usesConcept → Riemann–Roch theorem ⓘ

Chern character → usedIn → Riemann–Roch theorem ⓘ

this entity surface form: Riemann–Roch type formulas

Todd class → appearsIn → Riemann–Roch theorem ⓘ

this entity surface form: Riemann–Roch formulas

families index theorem → relatedTo → Riemann–Roch theorem ⓘ

this entity surface form: Riemann–Roch theorem for families

equivariant index theorem → relatedTo → Riemann–Roch theorem ⓘ