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:
“psychologist
Ultimately it’s all a matter of style. What it comes down to is this: Do you spell Jennifer with a J or G? That’s a class division. As a populist, I’m all for G.”
—Gore Vidal (b. 1925)
“Michelangelo said to Pope Julius II, “Self negation is noble, self-culture is beneficent, self-possession is manly, but to the truly great and inspiring soul they are poor and tame compared to self-abuse.” Mr. Brown, here, in one of his latest and most graceful poems refers to it in an eloquent line which is destined to live to the end of time—”None know it but to love it, None name it but to praise.””
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“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)