View PDF - Project Euclid
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CURVATURE AND COMPLEX SINGULARITIES<br />
487<br />
II kooOo,Oot (R) Zo.<br />
From the structure equations<br />
-<br />
dOo 0oo A 0o + 0o A 0<br />
0o, A 0, $,0oo,<br />
and we infer that<br />
6, 0, 8,0oo<br />
is the connection matrix for the Hermitian connection associated to the metric<br />
ds 0o0o<br />
on/f/. The curvature matrix is<br />
which, upon setting<br />
and using (4.29), gives<br />
0 003/ A 0o<br />
(4.30) 0o A 0o + 8 0 k,ro[coOor A 0o.<br />
As a check on signs and constants, the holomorphic sectional curvature in the<br />
direction Z1 is the coefficient of 0oa A 0o in qh, which by (4.30) is<br />
2 Z Ik1101 z,<br />
as it should be.<br />
The Chern forms of C N- are as usual defined by<br />
det(hS+ -1 )=<br />
27/"<br />
2 o<br />
Although we shall not need it, the quantities<br />
c()h_ g<br />
Cn () A<br />
are the coefficients in the expansion of the volume of the tube of radius r around<br />
inPN- . Also, by exact analogy with (4.17) it may be proved that (cf. 3(c))
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