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.
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 below), or for 1-dimensional projective spaces over many other fields.
See Chern-Simons for more discussion.
The Chern class 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.
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.
The Chern class in Chern-Weil theory
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.
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
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.,
This is evinced by the fact that a trivial bundle always admits a flat connection.
So, we shall show that
Consider the Kähler metric
One readily shows that the curvature 2-form is given by
Furthermore, by the definition of the first Chern class
We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere:
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 k-th Chern class of V, which is usually denoted ck(V), is an element of
the cohomology of X with integer coefficients. One can also define the total Chern class
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.
Classical axiomatic definition
The Chern classes satisfy the following four axioms:
Axiom 1. for all V.
Alexander Grothendieck axiomatic approach
- Functoriality: (Same as above)
- Additivity: If is an exact sequence of vector bundles, then .
- Normalization: If E is a line bundle, then where is the Euler class of the underlying real vector bundle.
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.
Namely, introducing the projectivization P(E) of the rank n complex vector bundle E → B as the fiber bundle on B whose fiber at any point is the projective space of the fiber Eb. The total space of this bundle P(E) is equipped with its tautological complex line bundle, that we denote τ, and the first Chern class
restricts on each fiber P(Eb) to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of complex projective spaces.
therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The Leray-Hirsch theorem then states that any class in H*(P(E)) can be written uniquely as a linear combination of the 1, a, a2, ..., an−1 with classes on the basis as coefficients.
In particular, one may define the Chern classes of E in the sense of Grothendieck, denoted by expanding this way the class , with the relation:
One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.
The top Chern class
In fact, these properties uniquely characterize the Chern classes. They imply, among other things:
- If n is the complex rank of V, then for all k > n. Thus the total Chern class terminates.
- The top Chern class of V (meaning , where n is the rank of V) is always equal to the Euler class of the underlying real vector bundle.
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
More generally, if is a direct sum of line bundles, with first Chern classes the Chern character is defined additively
Observe that when V is a sum of line bundles, the Chern classes of V can be expressed as elementary symmetric polynomials in the , In particular, on the one hand
while on the other hand
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
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
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:
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.
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 class in 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 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 classes of manifolds with structure
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.
Chern classes on arithmetic schemes and Diophantine equations
(See Arakelov geometry)
- Chern, S. S. (1946), "Characteristic classes of Hermitian Manifolds", Annals of Mathematics. Second Series (The Annals of Mathematics, Vol. 47, No. 1) 47 (1): 85–121, doi:10.2307/1969037, ISSN 0003-486X, JSTOR 1969037
- Grothendieck, Alexander (1958), "La théorie des classes de Chern", Bulletin de la Société Mathématique de France 86: 137–154, ISSN 0037-9484, MR 0116023
- Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-25907-7 (Provides a very short, introductory review of Chern classes).
- Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9