Chern character

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 found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

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.

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.

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 $H^2(X;\Z)$, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism):
$c_1(L \otimes L') = c_1(L) + c_1(L');$
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 hermitian vector bundle ''V'' of complex rank ''n'' over a smooth manifold ''M'', representatives of each Chern class (also called a Chern form) $c_k(V)$ of ''V'' are given as the coefficients of the characteristic polynomial of the curvature form $\Omega$ of ''V''.

$\det \left(\frac +I\right) = \sum_k c_k(V) t^k$

The determinant is over the ring of $n \times n$ matrices whose entries are polynomials in ''t'' with coefficients in the commutative algebra of even complex differential forms on ''M''. The curvature form $\Omega$ of ''V'' is defined as
\Omega = d\omega+\frac omega,\omega/math>
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 classes of the Chern forms do not depend on the choice of connection in ''V''.

If follows from the matrix identity $\mathrm(\ln(X))=\ln(\det(X))$ that $\det(X) =\exp(\mathrm(\ln(X)))$. Now applying the Maclaurin series for $\ln(X+I)$, we get the following expression for the Chern forms:

\sum_k c_k(V) t^k = \left I
+ i \frac t
+ \frac t^2
+ i \frac t^3
+ \cdots
\right

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 $\operatorname_n(\Complex)$ 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 $\pi\colon E \to B$ be a complex vector bundle over a paracompact space ''B''. Thinking of ''B'' as being embedded in ''E'' as the zero section, let $B' = E \setminus B$ and define the new vector bundle:
$E' \to B'$
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''.) Then $E'$ has rank one less than that of ''E''. From the Gysin sequence for the fiber bundle $\pi, _\colon B' \to B$:
$\cdots \to \operatorname^k(B; \Z) \overset \to \operatorname^k(B'; \Z) \to \cdots,$
we see that $\pi, _^*$ is an isomorphism for $k < 2n-1$. Let
$c_k(E) = \begin ^ c_k(E') & k < n\\ e(E_) & k = n \\ 0 & k > n \end$

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 $\mathbb^1$ be the Riemann sphere: 1-dimensional complex projective space. Suppose that ''z'' is a holomorphic local coordinate for the Riemann sphere. Let $V=T\mathbb^1$ be the bundle of complex tangent vectors having the form $a \partial/\partial z$ 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.,
$c_1(\mathbb^1\times \Complex)=0.$

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
$h = \frac.$

One readily shows that the curvature 2-form is given by
$\Omega=\frac.$

Furthermore, by the definition of the first Chern class
c_1= \left frac \operatorname \Omega\right.

We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere:
$\int c_1 =\frac\int \frac=2$
after switching to polar coordinates. By Stokes' theorem, an exact form would integrate to 0, so the cohomology class is nonzero.

This proves that $T\mathbb^1$ is not a trivial vector bundle.

Complex projective space

There is an exact sequence of sheaves/bundles:
$0 \to \mathcal_ \to \mathcal_(1)^ \to T\mathbb^n \to 0$
where $\mathcal_$ is the structure sheaf (i.e., the trivial line bundle), $\mathcal_(1)$ is Serre's twisting sheaf (i.e., the hyperplane bundle) 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 $c = 1 + c_1 + c_2 + \cdots$ (i.e., the Whitney sum formula),
$c(\Complex\mathbb^n) \overset= c(T\mathbb^n) = c(\mathcal_(1))^ = (1+a)^,$
where ''a'' is the canonical generator of the cohomology group $H^2(\Complex\mathbb^n, \Z )$; i.e., the negative of the first Chern class of the tautological line bundle $\mathcal_(-1)$ (note: $c_1(E^*) = -c_1(E)$ when $E^*$ is the dual of ''E''.)

In particular, for any $k\ge 0$,
$c_k(\Complex\mathbb^n) = \binom a^k.$

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:
$c_t(E) =1 + c_1(E) t + \cdots + c_n(E) t^n.$

This is not a new invariant: the formal variable ''t'' simply keeps track of the degree of ''c''_''k''(''E''). In particular, $c_t(E)$ is completely determined by the total Chern class of ''E'': $c(E) =1 + c_1(E) + \cdots + c_n(E)$ and conversely.

The Whitney sum formula, one of the axioms of Chern classes (see below), says that ''c''_''t'' is additive in the sense:
$c_t(E \oplus E') = c_t(E) c_t(E').$
Now, if $E = L_1 \oplus \cdots \oplus L_n$ is a direct sum of (complex) line bundles, then it follows from the sum formula that:
$c_t(E) = (1+a_1(E) t) \cdots (1+a_n(E) t)$
where $a_i(E) = c_1(L_i)$ are the first Chern classes. The roots $a_i(E)$, called the Chern roots of ''E'', determine the coefficients of the polynomial: i.e.,
$c_k(E) = \sigma_k(a_1(E), \ldots, a_n(E))$
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 $c_t(E)$ 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''
$t_1^k + \cdots + t_n^k = s_k(\sigma_1(t_1, \ldots, t_n), \ldots, \sigma_k(t_1, \ldots, t_n))$
with $s_1 = \sigma_1, s_2 = \sigma_1^2 - 2 \sigma_2$ and so on (cf. Newton's identities). The sum
$\operatorname(E) = e^ + \cdots + e^ = \sum s_k(c_1(E), \ldots, c_n(E)) / k!$
is called the Chern character of ''E'', whose first few terms are: (we drop ''E'' from writing.)
$\operatorname(E) = \operatorname + c_1 + \frac(c_1^2 - 2c_2) + \frac (c_1^3 - 3c_1c_2 + 3c_3) + \cdots.$

Example: The Todd class of ''E'' is given by:
$\operatorname(E) = \prod_1^n = 1 + c_1 + (c_1^2 + c_2) + \cdots.$

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. It is a classifying space in the sense that, given a complex vector bundle ''E'' of rank ''n'' over ''X'', there is a continuous map
$f_E: X \to G_n$
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:
f_E^*: \Z sigma_1, \ldots, \sigma_n\to H^*(X, \Z ).
One then puts:
$c_k(E) = f_E^*(\sigma_k).$

Remark: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let $\operatorname_n^$ 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 $\operatorname_n^ =$, G_n/math> to the cohomology functor $H^*(-, \Z ).$ 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'':
$\operatorname($, G_n H^*(-, \Z )) = H^*(G_n, \Z ) = \Z sigma_1, \ldots, \sigma_n

Computation formulae

Let ''E'' be a vector bundle of rank ''r'' and $c_t(E) = \sum_^r c_i(E)t^i$ the Chern polynomial of it.
*For the dual bundle $E^*$ of $E$, $c_i(E^*) = (-1)^i c_i(E)$.
*If ''L'' is a line bundle, then $c_t(E \otimes L) = \sum_^r c_i(E) c_t(L)^ t^i$ and so $c_i(E \otimes L), i = 1, 2, \dots, r$ are $c_1(E) + r c_1(L), \dots, \sum_^i \binom c_(E) c_1(L)^j, \dots, \sum_^r c_(E) c_1(L)^j.$
*For the Chern roots $\alpha_1, \dots, \alpha_r$ of $E$, $\begin c_t(\operatorname^p E) &= \prod_ (1 + (\alpha_ + \cdots + \alpha_)t), \\ c_t(\wedge^p E) &= \prod_ (1 + (\alpha_ + \cdots + \alpha_)t). \end$ In particular, $c_1(\wedge^r E) = c_1(E).$
*For example, for $c_i = c_i(E)$,
*:when $r = 2$, $c(\operatorname^2 E) = 1 + 3c_1 + 2 c_1^2 + 4 c_2 + 4 c_1 c_2,$
*:when $r = 3$, $c(\operatorname^2 E) = 1 + 4c_1 + 5 c_1^2 + 5 c_2 + 2 c_1^3 + 11 c_1 c_2 + 7 c_3.$
:(cf. Segre class#Example 2.)

Applications of formulae

We can use these abstract properties to compute the rest of the chern classes of line bundles on $\mathbb^1$. Recall that $\mathcal(-1)^* \cong \mathcal(1)$ showing $c_1(\mathcal(1)) = 1 \in H^2(\mathbb^1;\mathbb)$. Then using tensor powers, we can relate them to the chern classes of $c_1(\mathcal(n)) = n$ for any integer.

Properties

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 ''c_k''(''E''), is an element of
$H^(X;\Z),$
the cohomology of ''X'' with integer coefficients. O