On the value distribution of meromorphic mappings of covering spaces over \({\mathbb{C}}^ m\) into algebraic varieties
This paper is a succession of the author's previous papers [Hiroshima Math. J. 7, 833-853 (1977; Zbl 0412.32025); ibid. 10, 229-231 (1980; Zbl 0435.32017); Nagoya Math. J. 83, 213-233 (1981; Zbl 0429.32003)]. The problem is to establish the second main theorem of Nevanlinna and defect relations for meromorphic mappings between certain complex manifolds or spaces. Let \(\pi: X\to {\mathbb{C}}^ m\) be a finite (ramified) analytic covering space over \({\mathbb{C}}^ m\), V an n-dimensional complex projective manifold, D a reduced effective divisor on V and \(f: V\to V\) a meromorphic mapping such that \(f(X)\not\subset D.\) Assume that the closed image of the quasi-Albanese mapping \(\alpha: V-D\to A_{V-D}\) which is defined through logarithmic 1-forms along D, is of dimension n and of general type. Assume that f is non-degenerate with respect to \(| K_ V\otimes [D]|\); i.e., \(f(X)\not\subset Supp E\) for any \(E\in | K_ V\otimes [D]|\). The main aim is to prove the following inequality of the second main theorem type for f and D (main theorem (3.2)): There is \(K>0\) independent of f such that \(KT_ f(r)\leq N(r,Supp f^*D)+N(r,Supp R)+O(\delta \log r)+O(\log^+T_ f(r))\|_{E(\delta)},\) where \(\delta\in (0,1)\) and R denotes the ramification divisor of \(\pi: X\to {\mathbb{C}}^ m\) [the symbol \(''\|_{E(\delta)}''\) stands for that the stated (in)equality holds as r tends to infinity outside an exceptional subset E(\(\delta)\) of (0,\(\infty)\) which is a union of intervals with finite total measure and depends on \(\delta\) in this case]. The key of the proof of the main theorem (3.2) consists of a lemma on logarithmic derivatives given in terms of logarithmic 1-forms on V (section 1) and some investigations on the higher jet bundles over V combined with the quasi-Albanese mapping of V-D (section 2). - In the last section 4 we apply the main theorem (3.2) to obtain a multiplicity inequality for f and a finite family \(\{D_ i\}\) of divisors on V which extends Nevanlinna's ramification inequality, \(\sum^{q}_{i}1/\nu_ i\geq q-2\) for meromorphic functions on \({\mathbb{C}}\), where \(\nu\), \(1\leq i\leq q\) are positive integers such that \(f^*D_ i\geq \nu_ iSupp f^*D_ i\) for all i.
scientific article; zbMATH DE number 3903035