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364 19. Higher Dimensional Analogs of Elliptic Curves: Calabi–Yau Varieties

(5.6) Conjugation Invariant Polynomials in Matrix Elements. The ring of
GL(n, k)-conjugation invariant polynomials in the polynomial ring k[x 1,1 ,..., x n,n ]
2
on n -variables x i, j is k[c 1 (x), ..., c n (x)] where the polynomial c q (x) is defined by
the following GL(n, k)-invariant formula
q
R x (t) = det(I + [x i, j ]t) = c q (x)t .
0≤q≤n
In the case [x i, j ] is diagonal with x i, j = δ i, j λ i , we see that c q is an elementary
symmetric function

c q (x 1,1 ,..., x n,n ) = λ i(1) ...λ i(q) = s q (λ 1 ,...,λ n ).
i(1)<···<i(q)
Again we consider the logarithmic derivative

d q q
−t log(det(I + [x i, j ]t)) = Tr(X )(−t) .
dt
q≥1
Thus as in the case of symmetric functions, the subalgebra of GL(n, k)-conjugation
invariant functions has two forms
n
2
k[c 1 (x i, j ), ..., c n (x i, j )] = k[Tr(X), Tr(X ), ..., Tr(X )]
where X denotes the matrix [x i, j ]. The intersection of this subspace and the homo-
geneous polynomials of degree q is denoted by Inv q (n).

Let Inv(n) denote the direct sum over the homogeneous q Inv q (n).
(5.7) Definition. Let (E, ∇) be a pair consisting of a complex vector bundle E over
a smooth manifold M with connection ∇ having a curvature form & locally well
defined up to the inner automorphism, see (4.6). For any φ ∈ Inv(n),wehavea
well defined form φ(&) independent of the local coordinates of E. In particular, we
1
define the Chern forms c q (E, ∇) = c q (&) using the invariant polynomial c q
(2πi) q
introduced in (5.6). These Chern forms depend on the connection ∇ and the complex
vector bundle E.
(5.8) Remark. We can see that these forms φ(&) are closed by using the Bianchi
identity, see (4.6). In effect, d& = [ω, &] where ω is the corresponding connection
form related to ∇ for a local trivialization of E. Since it suffices to check that φ(&)
is closed on generators of

n
2
k[c 1 (x i, j ), ..., c n (x i, j )] = k[Tr(X), Tr(X ), ..., Tr(X )]
we calculate
q i j i j q
dTr(& ) = Tr(& (d&)& ) = Tr(& [ω, &]& ) = Tr([ω, & ]) = 0.
i+ j=q−1 i+ j=q−1