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456 PHILLIP A. GRIFFITHS
induced from that on ([N, and hence there is a unique Hermitian connection D
with the properties
D" g, and D is compatible with the metric. 1
The second property in (3.3) together with o + h 0 exactly imply that
{o} gives the connection matrix of the (pullback to o0(N) of) the universal
bundle.
The curvature matrix 12E {f} is, by the Cartan structure equation,
(3.4)
where we have used the second equation in (3.2). Setting
det (hi + X/-127r IE) = hn-kck(E)
defines the basic Chern forms C(OE), which are given explicitly by
(3.5)
A ,B
A,B
These are closed forms on G(n, N), and in de Rham cohomology they define the
Chern classes ck(E) H(G(n, N)). In particular the top Chern class c,(E) is
represented by
c,(f)
X/’
2-
We remark that under the obvious embedding
we have
(3.6) j, p f(dp)
det (ao).
j" G(n, N)----> GR(2n, 2N)
where Pf(dp) is the Pfaffian in the curvature matrix @ on G(2n, 2N) as defined
by (1.15). This is straightforward to verify from the definitions. ()
The formula (3.5) suggests that the Chern forms have sign properties in the
holomorphic case. For example, suppose that
f: S ---> G(2, n + 2)
is a holomorphic mapping of a complex surface into the Grassmannian, and