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.

Statements (47)

Predicate	Object
instanceOf	mathematical theorem → theorem →
appliesTo	compact Riemann surfaces → smooth projective algebraic curves →
completedBy	Gustav Roch NERFINISHED →
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 → 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 (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 NERFINISHED →
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 (5)

Subject (surface form when different)	Predicate
Atiyah–Singer index theorem →	generalizes
Riemann surface →	hasTheorem
Bernhard Riemann →	knownFor
Friedrich Bernhard Riemann →	notableConcept
Georg Friedrich Bernhard Riemann →	notableWork