Extremal holomorphic curves for defect relations
A holomorphic curve \(f=(f_0,\dots,f_n)\) is an element of the projective space \(\mathbb{P}^n\) with entire coordinates \(f_k\). Because we have \((f_0,\dots, f_n)=(gf_0, \dots,gf_n)\). One can find a representation of \(f\) with coordinates \(f_k\) having no common zeros. Below only such representations are considered. The Nevanlinna characteristic function of \(f\) is defined by \[ T(r,f)={1\over 2\pi} \int^{2\pi}_0 \ln\bigl\| f(re^{i\theta}) \bigr\| d\theta- \ln\bigl\| f(0)\bigr\| \] where \(\|(z_0, \dots, z_n)\| =\sqrt {| z_0|^2+ \cdots+ | z_n|^2}\). The value \(\rho=\varliminf_{s\to\infty} {\ln T(r,f)\over\ln r}\) is called a lower order of the holomorphic curve \(f\). Let \(a(w)=a(w_0,\dots,w_n)\) be a homogeneous polynomial. The set \(a=\{w\in\mathbb{P}^n: a(w)=0\}\) is called a hypersurface in \(\mathbb{P}^n\). Let \(N(r,a)\) be the Nevanlinna counting function of zeros of the entire function \(a\circ f\). A defect \(\delta(a,f)\) of a hypersurface \(a\) is defined by \(\delta(a,f)= \varliminf_{r\to\infty} \left(1-{N(r,a)\over dT(r,f)} \right)\) where \(d=\deg a(w)\). A system \(A\) of hypersurfaces \(a\) is called admissible if every system of \(n+1\) different polynomials \(a(w)\), \(a\in A\), have only \((0,\dots,0)\) as a common zero. In the book of S. Lang; Introduction to the complex hyperbolic spaces (1987; Zbl 0628.32001) it is proved that \(\sum_A\delta (a,f)\leq 2n\) for every admisible system \(A\). The author investigates the case of equality for curves of finite lower order \(\rho\). He proves: (1) \(2\rho\geq 2\) is an integer and \(\rho\) is an positive integer if \(\delta(a,f)=1\) for some hypersurface \(a\), (2) \(T(r,f)= r^\rho \ell(r)\) where \(\ell\) is a slowly varying function in the sense of Karamata, (3) \(\delta(a,f)= {p(a)\over \rho}\) where \(p(a)\in\mathbb{Z}_+\). The case \(n=1\) was considered by D. Drasin (1987).
scientific article; zbMATH DE number 1187477
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