Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants.
Chern classes were introduced by.

Geometric approach

Basic idea and motivation

Chern classes are characteristic classes. They are topological invariants associated with 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, the Chern classes can be expressed as polynomials in the coefficients of the curvature form.

Construction

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. For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space such that the bundle V is equal to the pullback, by f, 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. In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles.
It can be shown that for any two maps f, g from M to the classifying space whose pullbacks are the same bundle V, the maps must be homotopic. Therefore, the pullback by either f or g of any universal Chern class to a cohomology class of M must be the same class. This shows that the Chern classes of V are well-defined.
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. The resulting theory is known as the Chern–Weil theory.
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 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. Although that is strictly speaking a question about a real vector bundle, there are generalizations in which the hairs are complex, or for 1-dimensional projective spaces over many other fields.
See Chern–Simons theory 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, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism :
the tensor product of complex line bundles corresponds to the addition in the second cohomology group.
In algebraic geometry, this classification of complex line bundles by the first Chern class is a crude approximation to the classification 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.

Constructions

Via the Chern–Weil theory

Given a complex vector bundle V of complex rank n over a smooth manifold M, fix a vector bundle connection. Then representatives of each Chern class of V are given as the coefficients of the characteristic polynomial of the curvature form of.
The determinant is over the ring of matrices whose entries are polynomials in t with coefficients in the commutative algebra of even-degree complex differential forms on M. The curvature form 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 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. That is, Chern classes are cohomology classes in the sense of de Rham cohomology. It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection.
If follows from the matrix identity that. Now applying the Maclaurin series for, we get the following expression for the Chern forms:

Via an Euler class

One can define a Chern class in terms of an Euler class. 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 is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class and handles lower Chern classes in an inductive fashion.
The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let be a complex vector bundle over a paracompact space B. Thinking of B as being embedded in E as the zero section, let and define the new vector bundle:
such that each fiber is the quotient of a fiber F of E by the line spanned by a nonzero vector v in F Then has rank one less than that of E. From the Gysin sequence for the fiber bundle :
we see that is an isomorphism for. Let
It then takes some work to check the axioms of Chern classes are satisfied for this definition.
See also: The Thom isomorphism.

Examples

The complex tangent bundle of the Riemann sphere

Let be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann sphere. Let be the bundle of complex tangent vectors having the form 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:
after switching to polar coordinates. By Stokes' theorem, an exact form would integrate to 0, so the cohomology class is nonzero.
This proves that is not a trivial vector bundle.

Complex projective space

There is an exact sequence of sheaves/bundles:
where is the structure sheaf, is Serre's twisting sheaf and the last nonzero term is the tangent sheaf/bundle.
There are two ways to get the above sequence:
By the additivity of total Chern class ,
where a is the canonical generator of the cohomology group ; i.e., the negative of the first Chern class of the tautological line bundle
In particular, for any,

Chern polynomial

A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle E, the Chern polynomial c_t of E is given by:
This is not a new invariant: the formal variable t simply keeps track of the degree of c_k. In particular, is completely determined by the total Chern class of E: and conversely.
The Whitney sum formula, one of the axioms of Chern classes, says that c_t is additive in the sense:
Now, if is a direct sum of line bundles, then it follows from the sum formula that:
where are the first Chern classes. The roots, called the Chern roots of E, determine the coefficients of the polynomial: i.e.,
where σ_k are elementary symmetric polynomials. In other words, thinking of a_i as formal variables, c_k "are" σ_k. A basic fact on symmetric polynomials is that any symmetric polynomial in, say, t_i's is a polynomial in elementary symmetric polynomials in t_i's. Either by splitting principle or by ring theory, any Chern polynomial factorizes into linear factors after enlarging the cohomology ring; E need not be a direct sum of line bundles in the preceding discussion. The conclusion is
Example: We have polynomials s_k
with and so on. The sum
is called the Chern character of E, whose first few terms are:
Example: The Todd class of E is given by:
Remark: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let G_n be the infinite Grassmannian of n-dimensional complex vector spaces. This space is equipped with a tautologous vector bundle of rank, say. is called the classifying space for rank- vector bundles because given any complex vector bundle E of rank n over X, there is a continuous map
such that the pullback of to along is isomorphic to, and this map is unique up to homotopy. Borel's theorem says the cohomology ring of G_n is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σ_k; so, the pullback of f_E reads:
One then puts:
Remark: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let be the contravariant functor that, to a CW complex X, assigns the set of isomorphism classes of complex vector bundles of rank n over X and, to a map, its pullback. By definition, a characteristic class is a natural transformation from to the cohomology functor Characteristic classes form a ring because of the ring structure of cohomology ring. Yoneda's lemma says this ring of characteristic classes is exactly the cohomology ring of G_n: