There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.
The original approach to Chern classes was via algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case).
In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles.
There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case.
The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, locally free sheaves) over any nonsingular variety.
Algebro-geometric Chern classes do not require the underlying field to have any special properties.
Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem).
Although that is strictly speaking a question about a real vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.
The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking.
the tensor product of complex line bundles corresponds to the addition in the second cohomology group.
For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.
with ω the connection form and d the exterior derivative, or via the same expression in which ω is a gauge field for the gauge group of V. The scalar t is used here only as an indeterminate to generate the sum from the determinant, and I denotes the n × n identity matrix.
To say that the expression given is a representative of the Chern class indicates that 'class' here means up to addition of an exact differential form.
It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in V. If follows from the matrix identity
This is the approach in the book by Milnor and Stasheff, and emphasizes the role of an orientation of a vector bundle.
The basic observation is that a complex vector bundle comes with a canonical orientation, ultimately because
It then takes some work to check the axioms of Chern classes are satisfied for this definition.
We prove the complex version of the hairy ball theorem: V has no section which is everywhere nonzero.
By Stokes' theorem, an exact form would integrate to 0, so the cohomology class is nonzero.
is Serre's twisting sheaf (i.e., the hyperplane bundle) and the last nonzero term is the tangent sheaf/bundle.
The Whitney sum formula, one of the axioms of Chern classes (see below), says that ct is additive in the sense:
We can use these abstract properties to compute the rest of the chern classes of line bundles on
Given a complex vector bundle E over a topological space X, the Chern classes of E are a sequence of elements of the cohomology of X.
[clarification needed] The Chern classes satisfy the following four axioms: Alternatively, Alexander Grothendieck (1958) replaced these with a slightly smaller set of axioms: He shows using the Leray–Hirsch theorem that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.
It can be shown that there is a unique theory of Chern classes which assigns an algebraic vector bundle
Chern classes can be used to construct a homomorphism of rings from the topological K-theory of a space to (the completion of) its rational cohomology.
This last expression, justified by invoking the splitting principle, is taken as the definition ch(V) for arbitrary vector bundles V. If a connection is used to define the Chern classes when the base is a manifold (i.e., the Chern–Weil theory), then the explicit form of the Chern character is
) can be paired with the orientation homology class (or "integrated over the manifold") to give an integer, a Chern number of the vector bundle.
In algebraic geometry there is a similar theory of Chern classes of vector bundles.
If M is also compact and of dimension 2d, then each monomial of total degree 2d in the Chern classes can be paired with the fundamental class of M, giving an integer, a Chern number of M. If M′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M′ coincide with those of M. The theory also extends to real symplectic vector bundles, by the intermediation of compatible almost complex structures.