Page 236 - Curvature and Homology

2 1 8 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS
where the Qi are the forms 6* pulled down to M by means of the cross
section M + {(ala~j)~, (a/aZj)p). Denoting the components of the
torsion tensor by Tfli as in 5 5.3, put

then, f is a real-valued function.

Lemma 6.8.1, The Tjki are the constants of structure of a local Lie group.
For,
- *Sdf = gT8*D, D, f

Hence, since the curvature is zero, an application of the interchange
formula (1.7.2 1) gives
- $Sdf = grl' D T i D Tjk
r jk s i'
Therefore, by proposition 6.5.1,
Af = Sdf r 0,

from which we conclude that the D,Tjki vanish. Consequently, from
(5.3.22) they satisfy the Jacobi identities

Since M is complex parallelisable, it follows from the proof of
theorem 6.7.3 that there exists n linearly independent holomorphic
pfaffian forms 01, .a*, 8n defined everywhere on M. Therefore, their
exterior products 8" A Oj (i < j) are also holomorphic and linearly
independent everywhere (cf. lemma (6.10.1)). Moreover, since there are
n(n - 1)/2 such products they form a basis of the space of pure forms
of bidegree (2,O).
It is now shown that d8i is a holomorphic 2-form, i = 1, ..-, n. Indeed,
8i is of bidegree (1,0), and so since d8i = d'Oi (by virtue of the fact that
the 8i are holomorphic), d8i is a pure form of bidegree (2,O). On the
other hand, dud@ = d"d'8i = 0 since d'd" + d"d' = 0.
We conclude that the d8i may be expressed linearly (with complex
coefficients) in terms of the products Oj A Ok, and since M is compact
these coefficients (as holomorphic functions) are necessarily constants.
That the coefficients are proportional to the Tjki is easily seen from