Abstract Suppose that M is a parabolic manifold of dimension m and V is a vector space of dimension n + 1 1. Let / : M P(V) be a meromorphic map and (5 be a finite set of meromorphic maps g : M P(V*) in general position. Also, suppose that each g in 0 grows slower than / . Then, the following defect relation can be proven: X) «(/,/) n+1. see This is derived using the original method of Cartan [3] and following the ideas of the Lemma of the Logarithmic Derivative over C m given by Vitter [17] in 1977. In order to achieve this result, the mild assumption must be made that there exists a holomorphic m form G o n M . This is equivalent to assuming that there exists a nontrivial holomorphic section of the canonical bundle K of M. Also, the Ricci curvature of the manifold M and the norm of G taken along the fibres of K play key roles in the derivation the Lemma of the Logarithmic Derivative in this case. This result holds on all parabolic, Stein manifolds. In particular, if Mi is a noncompact, parabolic Riemann surface for 1 i m, then this relation is valid on Mi x ... x Mm. Also, if n : M N is a holomorphic line bundle where N a connected, compact complex manifold of dimension m 1 0, then M is another example of a parabolic manifold on which the defect relation holds under the standard assumptions. vm
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