Chapter 1 1.1 Parabolic manifolds and the Ricci functions The defect relation for moving targets has been sought after since the late 1920's. Chuang [4] provided a first step in this direction by finding a defect relation for entire holomorphic functions of a single complex variable. Good results have been obtained utilizing the methods of Steinmetz [8] and Ru-Stoll [7] which follow the proofs of the defect relations of Nevanlinna [5] and Cartan [3] in the fixed target case. The defect relations proven by Stoll using the Ahlfors-Weyl theory require undesirable assumptions in the moving target case. The Steinmetz technique as well as the Cartan proof require a Lemma of the Logarithmic Derivative which was proven for C by Nevanlinna [5], for C n by Vitter [17], and for parabolic cov- ering manifolds by Bardis [2]. Now, a Lemma of the Logarithmic Derivative on a parabolic manifold M is proven under the mild assumption that the canonical bundle KM of M has a non-trivial holomorphic section. This lemma is then used to prove the defect relation for moving targets on M. To begin with, assume that M is a connected, complex manifold of dimension m. The notion of a parabolic exhaustion must be introduced. A proper, C°° map r : M R+ is called an exhaustion of M. For r 0 and S C M, define v = ddcr LO = ddc log T a = dclogr Acj m _ 1 5+ = {xe S\v(x) 0} S* = {x e S\T{X) 0} S r = {xe S\r(x) = r 2 } S(r) = {x e S\T(X) r 2 } S[r] = {xe S\r{x) r 2 } . If UJ 0 and VTH 0 = UJ171 = da, then the exhaustion r is called parabolic. As a result, v 0. If v 0, then r is called strictly parabolic. If r is a parabolic (resp. strictly parabolic) exhaustion for M, then (M, r) is called a parabolic (resp. strictly parabolic) manifold. 3
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