The Chern Class of Line Bundles
See also: Exponential sheaf sequencefor a sheaf-theoretic description.
An important special case occurs when V is a line bundle. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Euler class of the bundle.
The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking. That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of H²(X;Z), which associates to a line bundle its first Chern class. Addition in the second dimensional cohomology group coincides with tensor product of complex line bundles.
In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) holomorphic line bundles by linear equivalence classes of divisors.
For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.
Read more about this topic: Chern Class
Famous quotes containing the words class, line and/or bundles:
“To avoid the consequences of posterity the mulattos give the blacks a first class letting alone. There is a frantic stampede white-ward to escape from Jamaica’s black mass.”
—Zora Neale Hurston (1891–1960)
“There’s something like a line of gold thread running through a man’s words when he talks to his daughter, and gradually over the years it gets to be long enough for you pick up in your hands and weave into a cloth that feels like love itself. It’s another thing, though, to hold up that cloth for inspection.”
—John Gregory Brown (20th century)
“He bundles every forkful in its place,
And tags and numbers it for future reference,
So he can find and easily dislodge it
In the unloading. Silas does that well.
He takes it out in bunches like birds’ nests.”
—Robert Frost (1874–1963)