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
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