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 (and some types of algebraic 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 (an infinite Grassmannian in this case). 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 (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 theory for more discussion.
Abstract definition
Given a complex vector bundle E over a topological space X, the Chern classes of E are a sequence of elements of the cohomology of X. The k-th Chern class of E, which is usually denoted ck(E), 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: for all E. Naturality: If is continuous and f*E is the vector bundle pullback of E, then .Milnor & Stasheff 74, Lemma 14.2Hatcher 17, Thrm. 3.2 Whitney sum formula: If is another complex vector bundle, then the Chern classes of the direct sum are given byMilnor & Stasheff 74, Equation (14.7)Lawson & Michelson 90, Equation (B.9) that is, Normalization: The total Chern class of the tautological line bundle over is 1−H, where H is Poincaré dual to the hyperplane .
Grothendieck axiomatic approach
Alternatively, replaced these with a slightly smaller set of axioms:
Naturality: (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 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 is equipped with its tautological complex line bundle, that we denote , and the first Chern class
restricts on each fiber 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.
The classes
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 can be written uniquely as a linear combination of the 1, a, a2, ..., an−1 with classes on the base 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 class of line bundles
(Let X be a topological space having the homotopy type of a CW complex.)
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.Milnor & Stasheff 74; Chapter 14, Definition on p. 158
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 (thus an isomorphism):
the tensor product of complex line bundles corresponds to the addition in the second cohomology group.
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.
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 (also called a Chern form) 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 .
It 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 (the Euler class of the underlying real vector bundle) 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 (a point of B′ is specified by a fiber F of E and a nonzero vector on F.)Editorial note: Our notation differs from Milnor−Stasheff, but seems more natural. 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:The sequence is sometimes called the Euler sequence.
where is the structure sheaf (i.e., the trivial line bundle), is Serre's twisting sheaf (i.e., the hyperplane bundle) and the last nonzero term is the tangent sheaf/bundle.