Chern class |
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes (pronounced chen ) are a particular type of characteristic class associated to complex vector bundles.
Chern classes are named for Shiing-Shen Chern, who first gave a general definition of them in the 1940s.
=Example: Riemannian manifolds=
Given a complex vector bundle V over a Riemannian manifold M , a representative of each Chern class c_k(V) of V is given as the product of the eigenvalues of the field strength F of the vector bundle:
:det left(frac {itF}{2pi} -I ight) = sum_{k=0}^n c_k(V) t^k
Here, F is the field strength or curvature form of the bundle, which is given locally (on each chart) as
:F=dA
with A the connection (mathematics) and d the exterior derivative. ( F is only locally an exact form, and not globally, as the vector bundle may have a non-trivial topology.) The rank (or complex dimension)of a fibre of the vector bundle is n . The dimensionality of the fibres need have no relation to the dimension of the underlying manifold. The scalar t is used here only as a real number, to generating function the sum from the determinant. Here, I is the n × n identity matrix.
To understand this expression, recall that F is a 2-form on the manifold M taking values that are complex n × n matrices. For any given, fixed matrix n × n complex matrix B , one may construct a set of homogeneous polynomial symmetric polynomials P_k of degree k by making the expansion:
:det left(I +tB ight) = sum_{k=0}^n t^k P_k(B)
It should be clear that each P_k is just a polynomial in the eigenvalues of B . These polynomials are invariant under a change of basis: that is, given phi in GL(n,mathbb{C}), one has
:P_k(phi B phi^{-1}) = P_k(B)
This later consideration shows that the Chern classes are invariant under gauge transformations as well as changes of coordinate charts; in either case, the curvature tensor F transforms as
:F o phi F phi^{-1}
with phi taken from the gauge group of the vector bundle.
The field strength F may be viewed as a matrix of 2-forms, and thus the eigenvalues of F are 2-forms as well. Each Chern class is then a 2k form.
To be precise, the above only defined a representative element of a Chern class. The other representative elements are those that are cohomologous to it; that is, each Chern class is defined to be a member of the cohomology group H^{2k}(V). By varying the connection, other representatives are obtained. This essentially follows from the Bianchi identity on the field strength:
:DF = 0
which implies that for each symmetric polynomial P_k(F), on has
:dP_k(F) = 0
The above leads to the idea that different choices for the connection only lead to representatives c_k(V) that are cohomologous to each other.
The first two Chern classes may be written explicitly as
:c_1(V) = frac{i}{2pi} operatorname {tr} F
and
:c_2(V) = frac{n-1}{2n} c_1(V) wedge c_1(V) + frac{1}{8pi^2} operatorname{tr} (F_z wedge F_z)
where F_z is the trace-free part of F :
:F_z = F - frac{I}{n} operatorname{tr} F
and I in the 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. That is, Chern classes are cohomology classes in the sense of de Rham cohomology.
= Properties of Chern classes =
Given a complex vector bundle V over a topological space X , the Chern classes of V are a sequence of elements of the Cohomology of X . The kth Chern class of V , which is usually denoted ck ( V ), is an element of
: H2k ( X ;Z),
the cohomology of X with integer coefficients. One can also define the total Chern class
:c(V) = c_0(V) + c_1(V) + c_2(V) + cdots .
The Chern classes satisfy the following four properties.
:c(V oplus W) = c(V) cup c(W);
that is,
:c_k(V oplus W) = sum_{i = 0}^n c_i(V) cup c_{k - i}(W).
In fact, these properties uniquely characterize the Chern classes. They imply, among other things:
Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.
= Construction of Chern classes =
There are various ways of approaching the subject. Originally Chern used differential geometry. In algebraic topology the Chern classes arise via homotopy theory which provides a mapping associated to V to a classifying space (an infinitary Grassmannian in this case). There is an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case. Chern classes also arise naturally in algebraic geometry.
The intuitive meaning of the Chern class concerns required zeroes of a ), though that is strictly speaking a question about a real vector bundle.
See Chern-Simons for more discussion.
= Chern classes of line bundles =
If V is a line bundle there is just a single (first) Chern class in the second cohomology group of X . 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 H 2( 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.
= Chern classes of almost complex manifolds and cobordism =
The theory of Chern classes gives rise to cobordism invariants for almost complex manifolds.
If M is an almost complex manifold, then its tangent bundle is a complex vector bundle. The Chern classes of M are thus defined to be the Chern classes of its tangent bundle. If M is also compact and of dimension 2 d , then each Monomial of total degree 2 d 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 .
= Generalizations =
There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a generalized cohomology theory. The theories for which such generalization is possible are called complex orientable . The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a formal group law.
=See also=
=References=
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