Chern classes

E240803

characteristic class cohomology class mathematical concept topological invariant

Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.

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All labels observed (2)

Label	Occurrences
Chern classes canonical	4
Bott–Chern classes	1

Statements (50)

Predicate	Object
instanceOf	characteristic class ⓘ cohomology class ⓘ mathematical concept ⓘ topological invariant ⓘ
appliesTo	complex line bundle ⓘ complex vector bundle ⓘ holomorphic vector bundle ⓘ
captures	curvature information of complex vector bundles ⓘ twisting of complex vector bundles ⓘ
component	first Chern class ⓘ higher Chern classes ⓘ second Chern class ⓘ total Chern class ⓘ
definedOn	base space of a vector bundle ⓘ
field	algebraic geometry ⓘ differential geometry ⓘ topology ⓘ
introducedBy	Shiing-Shen Chern ⓘ
namedAfter	Shiing-Shen Chern ⓘ
notation	c(E) ⓘ c_1(E) ⓘ c_2(E) ⓘ c_i(E) ⓘ
property	are additive under direct sum of bundles ⓘ are functorial ⓘ are multiplicative under tensor product in total Chern class form ⓘ are natural with respect to pullback of bundles ⓘ are stable under isomorphism of bundles ⓘ satisfy Whitney sum formula ⓘ vanish above the rank of the bundle ⓘ
relatedTo	Chern character ⓘ K-theory ⓘ Pontryagin classes ⓘ Stiefel–Whitney classes ⓘ Todd class ⓘ
takesValuesIn	Chow ring ⓘ integral cohomology ⓘ singular cohomology ⓘ
timePeriod	mid 20th century ⓘ
usedFor	Chern–Weil theory ⓘ surface form: Gauss–Bonnet–Chern theorem Grothendieck–Riemann–Roch theorem ⓘ Hirzebruch–Riemann–Roch theorem ⓘ Riemann–Roch theorem ⓘ surface form: Riemann–Roch theorems classification of complex line bundles via first Chern class ⓘ classification of complex vector bundles up to isomorphism ⓘ computation of characteristic numbers ⓘ definition of Chern character ⓘ index theorems ⓘ intersection theory on algebraic varieties ⓘ obstruction theory ⓘ

Referenced by (5)

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

Shiing-Shen Chern → knownFor → Chern classes ⓘ

Grothendieck–Riemann–Roch theorem → requires → Chern classes ⓘ

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

Raoul Bott → knownFor → Chern classes ⓘ

this entity surface form: Bott–Chern classes

Todd class → constructedFrom → Chern classes ⓘ