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 ct 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
ck(
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
ct 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
ai as formal variables,
ck "are" σ
k. A basic fact on
symmetric polynomials is that any symmetric polynomial in, say, ''t'
is is a polynomial in elementary symmetric polynomials in ''t'
is. 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
sk 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{ch}(E) = e^{a_1(E)} + \cdots + e^{a_n(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{ch}(E) = \operatorname{rk} + c_1 + \frac{1}{2}(c_1^2 - 2c_2) + \frac{1}{6} (c_1^3 - 3c_1c_2 + 3c_3) + \cdots.
Example: The
Todd class of
E is given by:\operatorname{td}(E) = \prod_1^n {a_i \over 1 - e^{-a_i}} = 1 + {1 \over 2} c_1 + {1 \over 12} (c_1^2 + c_2) + \frac{1}{24}c_1c_2\cdots.
Remark: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let
Gn be the
infinite Grassmannian of
n-dimensional complex vector spaces. This space is equipped with a tautologous vector bundle of rank n, say E_n \to G_n. G_n is called the
classifying space for rank-n vector bundles because given any complex vector bundle
E of rank
n over
X, there is a continuous map f_E: X \to G_n such that the pullback of E_n to X along f_E is isomorphic to E, and this map f_E is unique up to homotopy.
Borel's theorem says the cohomology ring of
Gn is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σ
k; so, the pullback of
fE 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{Vect}_n^{\Complex} 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{Vect}_n^{\Complex } = [-, G_n] 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
Gn: \operatorname{Nat}([-, G_n], H^*(-, \Z )) = H^*(G_n, \Z ) = \Z [\sigma_1, \ldots, \sigma_n]. == Computation formulae ==