Chern Class - The Chern Class of Line Bundles

The Chern Class of Line Bundles

See also: Exponential sheaf sequence

for 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:

    “But you must know the class of sweet women—who are always so happy to declare “they have all the rights they want”; “they are perfectly willing to let their husbands vote for them”Mare and always have been numerous, though it is an occasion for thankfulness that they are becoming less so.”
    —Eliza “Mother” Stewart (1816–1908)

    “It is a great many years since at the outset of my career I had to think seriously what life had to offer that was worth having. I came to the conclusion that the chief good for me was freedom to learn, think, and say what I pleased, when I pleased. I have acted on that conviction... and though strongly, and perhaps wisely, warned that I should probably come to grief, I am entirely satisfied with the results of the line of action I have adopted.”
    —Thomas Henry Huxley (1825–95)

    “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)