Chern class

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Chern class

In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.

Chern classes were introduced by Shiing-Shen Chern (1946).

Basic idea and motivation

Chern classes are characteristic classes. They are topological invariants associated to vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.

In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sections a vector bundle has. The Chern classes offer some information about this through, for instance, the Riemann-Roch theorem and the Atiyah-Singer index theorem.

Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form.

The Chern class of a Hermitian vector bundle on a smooth manifold

Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M, a representative of each Chern class (also called a Chern form) ck(V) of V are given as the coefficients of the characteristic polynomial of the curvature form Ω of V.

$\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k$

The determinant is over the ring of n×n matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form Ω of V is defined as

$\Omega=d\omega+\frac{1}{2}[\omega,\omega]$

with ω the connection form and d the exterior derivative, or via the same expression in which ω is a gauge form 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. That is, Chern classes are cohomology classes in the sense of de Rham cohomology. It can be shown that the cohomology class of the Chern forms do not depend on the choice of connection in V.

Example: the complex tangent bundle of the Riemann sphere

Let CP1 be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann sphere. Let V = TCP1 be the bundle of complex tangent vectors having the form a∂/∂z at each point, where a is a complex number. We prove the complex version of the hairy ball theorem: V has no section which is everywhere nonzero.

For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e.,

$c_1({\mathbf C\mathbf P}^1\times {\mathbf C})=0.$

This is evinced by the fact that a trivial bundle always admits a flat connection.

So, we shall show that

$c_1(V) \not= 0.$

Consider the Kähler metric

$h = \frac{dzd\bar{z}}{(1+|z|^2)^2}.$

One readily shows that the curvature 2-form is given by

$\Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^2)^2}.$

Furthermore, by the definition of the first Chern class

$c_1=\frac{i}{2\pi}\Omega.$

We must show that the cohomology class of this is non-zero. It suffices to compute its integral over the Riemann sphere:

$\int c_1=\frac{i}{\pi}\int \frac{dz\wedge d\bar{z}}{(1+|z|^2)^2}=2$

after switching to polar coordinates. By Stokes' theorem, an exact form would integrate to 0, so the cohomology class is nonzero.

This proves that TCP1 is not a trivial vector bundle.

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 axioms:

Axiom 1. c0(V) = 1 for all V.

Axiom 2. Functoriality: If $f : Y \to X$ is continuous and f * V is the vector bundle pullback of V, then ck(f * V) = f * ck(V).

Axiom 3. Whitney sum formula: If $W \to X$ is another complex vector bundle, then the Chern classes of the direct sum $V \oplus W$ are given by

$c(V \oplus W) = c(V) \smile c(W);$

that is,

$c_k(V \oplus W) = \sum_{i = 0}^k c_i(V) \smile c_{k - i}(W).$

Axiom 4. Normalization: The total Chern class of the tautological line bundle over $\mathbf{CP}^k$ is 1 − H, where H is Poincaré-dual to the hyperplane $\mathbf{CP}^{k - 1} \subseteq \mathbf{CP}^k$.

Alternatively, Alexander Grothendieck replaced these with a slightly smaller set of axioms:

• Functoriality: (Same as above)
• Additivity: If $\ 0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$ is an exact sequence of vector bundles, then $c(E)=c(E')\smile c(E'')$.
• Normalization: If E is a line bundle, then $c(E)=1+e(E_{\mathbf R})$ where $e(E_{\mathbf R})$ is the Euler class of the underlying real vector bundle.

In fact, these properties uniquely characterize the Chern classes. They imply, among other things:

• If n is the complex rank of V, then ck(V) = 0 for all k > n. Thus the total Chern class terminates.
• The top Chern class of V (meaning cn(V), where n is the rank of V) is always equal to the Euler class of the underlying real vector bundle.

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, 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 to V to a classifying space (an infinite Grassmannian in this case). Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.

Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his.

There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case.

Chern classes arise naturally in algebraic geometry. 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. In particular, the vector bundles need not necessarily be complex.

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 above), or for 1-dimensional projective spaces over many other fields.

See Chern-Simons for more discussion.

Chern classes of line bundles

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.[citation needed]

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 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.

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.

Chern numbers

If we work on an oriented manifold of dimension 2n, then any product of Chern classes of total degree 2n 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 c1³, c1c2, and c3. In general, if the manifold has dimension 2n, the number of possible independent Chern numbers is the number of partitions of n.

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.

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

$\operatorname{ch}(L) = \exp(c_{1}(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.$

More generally, if $V = L_1 \oplus ... \oplus L_k$ is a direct sum of line bundles, with first Chern classes xi = c1(Li), the Chern character is defined additively

$\operatorname{ch}(V) = e^{x_1} + \dots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + ... + x_k^m).$

Observe that when V is a sum of line bundles, the Chern classes of V can be expressed as elementary symmetric polynomials in the xi, ci(V) = ei(x1,...,xn). In particular, on the one hand

$c(V) := \sum_{i=0}^n c_i(V),$

while on the other hand

$c(V) = c(L_1 \oplus \dots \oplus L_n) = \prod_{i=1}^n c(L_i) = \prod_{i=1}^n (1+x_i) = \sum_{i=0}^n e_i(x_1,\dots,x_n).$

Consequently, Newton's identities may be used to reexpress the power sums in ch(V) above solely in terms of the Chern classes of V, giving

$\operatorname{ch}(V) = \operatorname{dim}(V) + c_1(V) + \frac{1}{2}(c_1(V)^2 - 2c_2(V)) + \frac{1}{6} (c_1(V)^3 - 3c_1(V)c_2(V) + 3c_3(V)) + ...,$

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, then the explicit form of the Chern character is

$\hbox{ch}(V)=\hbox{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)$

where Ω is the 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:

$\hbox{ch}(V\oplus W)=\hbox{ch}(V)+\hbox{ch}(W)$
$\hbox{ch}(V\otimes W)=\hbox{ch}(V)\hbox{ch}(W).$

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 groups 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 classes on arithmetic schemes and Diophantine equations

(See Arakelov geometry)

References

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