Editing Chern class (section)

==Proximate notions==

===The Chern character===
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. For a line bundle ''L'', the Chern character ch is defined by

<math display="block">\operatorname{ch}(L) = \exp(c_1(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.</math>

More generally, if <math>V = L_1 \oplus \cdots \oplus L_n</math> is a direct sum of line bundles, with first Chern classes <math>x_i = c_1(L_i),</math> the Chern character is defined additively
<math display="block"> \operatorname{ch}(V)  = e^{x_1} + \cdots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + \cdots + x_n^m). </math>

This can be rewritten as:<ref>(See also {{slink||Chern polynomial}}.) Observe that when ''V'' is a sum of line bundles, the Chern classes of ''V'' can be expressed as [[elementary symmetric polynomials]] in the <math>x_i</math>, <math>c_i(V) = e_i(x_1,\ldots,x_n).</math>
In particular, on the one hand
<math display="block">c(V) := \sum_{i=0}^n c_i(V),</math>
while on the other hand
<math display="block">\begin{align}
c(V) &= c(L_1 \oplus \cdots \oplus L_n) \\
&= \prod_{i=1}^n c(L_i) \\
&= \prod_{i=1}^n (1+x_i) \\
&= \sum_{i=0}^n e_i(x_1,\ldots,x_n)
\end{align}</math>
Consequently, [[Newton's identities#Expressing power sums in terms of elementary symmetric polynomials|Newton's identities]] may be used to re-express the power sums in ch(''V'') above solely in terms of the Chern classes of ''V'', giving the claimed formula.</ref>

<math display="block"> \operatorname{ch}(V) = \operatorname{rk}(V) + c_1(V) + \frac{1}{2}(c_1(V)^2 - 2c_2(V)) + \frac{1}{6} (c_1(V)^3 - 3c_1(V)c_2(V) + 3c_3(V)) + \cdots.</math>

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
<math display="block">\operatorname{ch}(V)=\left[\operatorname{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)\right]</math>
where {{math|Ω}} is the [[curvature form|curvature]] of the connection.

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product.  Specifically, it obeys the following identities:

<math display="block">\operatorname{ch}(V \oplus W) = \operatorname{ch}(V) + \operatorname{ch}(W)</math>
<math display="block">\operatorname{ch}(V \otimes W) = \operatorname{ch}(V) \operatorname{ch}(W).</math>

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ''ch'' is a [[homomorphism]] of [[abelian group]]s from the [[K-theory]] ''K''(''X'') into the rational cohomology of ''X''.  The second identity establishes the fact that this homomorphism also respects products in ''K''(''X''), and so ''ch'' is a homomorphism of rings.

The Chern character is used in the [[Hirzebruch–Riemann–Roch theorem]].

===Chern numbers===

If we work on an [[orientable manifold|oriented manifold]] of dimension <math>2n</math>, then any product of Chern classes of total degree <math>2n</math> (i.e., the sum of indices of the Chern classes in the product should be <math>n</math>) can be paired with the [[orientation homology class]] (or "integrated over the manifold") to give an integer, a '''Chern number''' of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by <math>c_1^3</math>, <math>c_1 c_2</math>, and <math>c_3</math>.  In general, if the manifold has dimension <math>2n</math>, the number of possible independent Chern numbers is the number of [[integer partition|partition]]s of <math>n</math>.

The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.

===Generalized cohomology theories===

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 cobordism#Formal group laws|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]].

===Algebraic geometry===

In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in:

*For complex varieties the Chern classes can take values in ordinary cohomology, as above.
*For varieties over general fields, the Chern classes can take values in cohomology theories such as [[etale cohomology]] or [[l-adic cohomology]].
*For varieties ''V'' over general fields the Chern classes can also take values in homomorphisms of [[Chow group]]s CH(V): for example, the first Chern class of a line bundle over a variety ''V'' is a homomorphism from CH(''V'') to CH(''V'') reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the [[cap product]].

=== Manifolds with structure ===

The theory of Chern classes gives rise to [[cobordism]] invariants for [[almost complex manifold]]s.

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 space|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''.

The theory also extends to real [[Symplectic geometry|symplectic]] vector bundles, by the intermediation of compatible almost complex structures. In particular, [[symplectic manifold]]s have a well-defined Chern class.

=== Arithmetic schemes and Diophantine equations ===

(See [[Arakelov geometry]])