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Holomorphic curves into algebraic varieties. (English) Zbl 1184.32006
The paper under review deals with a defect relation for holomorphic curves from \(\mathbb{C}\) into smooth complex projective algebraic varieties, and the author extends his previous result [Am. J. Math. 126, No. 1, 215–226 (2004; Zbl 1044.32009)]. Let \(V\subseteq \mathbb{P}^N(\mathbb{C})\) be a smooth complex projective variety of dimension \(n\). Let \(D_j\) \((j= 1,\dots,q)\) be hypersurfaces of degree \(d\), in \(\mathbb{P}^N(\mathbb{C})\), where \(q> n\). Suppose that \(V\cap\text{Supp\,}D_{j_0}\cap\cdots\cap \text{Supp\,}D_{j_n}= \emptyset\) for each subset \(\{j_0,\dots, j_n\}\subset\{1,\dots, q\}\). Let \(f:\mathbb{C}\to V\) be an algebraically nondegenerate holomorphic curve and denote by \(T_f(r)\) the characteristic function of \(f\) with respect to the hyperplane bundle \(L(H)\) over \(\mathbb{P}^N(\mathbb{C})\).
Then the main theorem in this paper is as follows: For an arbitrary positive real number \(\varepsilon\), the inequality \(\sum^q_{j=1} d^{-1}_j m_f(r, D_n)\leq (n+ 1+\varepsilon) T_f(r)\) holds for all \(r\in\mathbb{R}^+\) outside a Borel subset with finite Lebesgue measure. From this theorem follows the defect relation \[ \sum^q_{j=1} \delta_f(D_j)\leq \dim V+ 1. \] The method of proof of the main theorem is strongly motivated by the analogy between Nevanlinna theory and Diophantine approximation.
In the proof, the author uses the results due to J.-H. Evertse and R. G. Ferretti [in: Schlickewei, Hans Peter et al., Diophantine approximation. Festschrift for Wolfgang Schmidt. Based on lectures given at a conference at the Erwin Schrödinger Institute, Vienna, Austria, 2003. Wien: Springer. Developments in Mathematics 16, 175–198 (2008; Zbl 1153.11032)]. The main theorem in this paper can be considered as the analytic counterpart of Corollary 1.2 of Evertse-Ferretti’s paper.

MSC:
32H25 Picard-type theorems and generalizations for several complex variables
32H30 Value distribution theory in higher dimensions
11K60 Diophantine approximation in probabilistic number theory
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