Page 249 - Curvature and Homology

any order provided the Ricci curvature defined by the given differentials
is negative. In fact, the condition that the differentials be df-closed
ensured the existence of a Kaehler metric relative to which the Ricci
curvature was non-positive. By restricting the independence assumption
on the holomorphic differentials we may drop the restriction on the
curvature entirely, thereby obtaining interesting consequences from an
algebraic point of view.
We consider a compact complex manifold M of complex dimension n.
No assumption regarding a metric will be made, that is, in particular,
M need not be a Kaehler manifold. Let a be a holomorphic form of
bidegree (1,O) and X a holomorphic (contravariant) vector field on M.
Then, since M is compact
i(X)a = const.,

for, i(X)a is a holomorphic function on M. If we assume that there are
N > n holomorphic 1-forms al, ..., aN defined on M, then

where the cr, r = 1, --., N are constants. If, for any system of constants
cr (not all zero) the linear equations (6.13.1) are independent, that is,
if the rank of the matrix

is n + 1 at some point, the holomorphic vector field X must vanish.
Now, let t be a holomorphic contravariant tensor field of order p on M.
Then, under the conditions, the same conclusion prevails, that is, t must
vanish. Indeed, it is known for p = 1. Applying induction, assume the
validity of the statement for holomorphic contravariant tensor fields of
order p - 1 and consider the holomorphic tensor field

are the components of N holomorphic contravariant tensor fields of
order p - 1. By the inductive assumption they must vanish. But, we
have assumed that at least n of the differentials ar are independent. Thus,
the coefficients of the a(;: in the system of linear equations

must vanish.