Finiteness theorems for meromorphic mappings (Q1268680)

The main purpose of this paper is to prove finiteness theorems for some families of meromorphic mappings that are transcendental in general. Let \(L\to M\) be a fixed line bundle over \(M,\) and let \(\sigma_1,\dots,\sigma_s\) be linearly independent holomorphic sections of \(L \to M\) with \(s\geq 2.\) It is assumed that \((\sigma_j) = dD_j\) \((1 \leq j \leq s)\) for some positive integer \(d,\) where \(D_j\) are effective divisors on \(M.\) Set \(\varpi =c_1\sigma_1 +\dots+c_s\sigma_s,\) where \(c_j\in{\mathbb C}^\ast.\) Let \(D\) be a divisor defined by \(\varpi = 0.\) A meromorphic mapping \(\varPsi: M \to{\mathbb P}_{s-1}({\mathbb C})\) is defined by \(\varPsi=(\sigma_1,\dots,\sigma_s).\) Let \(p\) be a nonnegative integer. For a nonzero effective divisor \(E\) on \(\mathbb C^m,\) \(\mathcal F(p;(\mathbb C^m,E),(M,D))\) denotes the set of all meromorphic mappings \(f:\mathbb C^m\to M\) such that \(f^\ast D\equiv E\pmod p\). A meromorphic mapping \(f:\mathbb C^m\to M\) is said to be analytically nondegenerate if \(f(\mathbb C^m)\) is not included in any proper analytic subset of \(M.\) Let \(\mathcal F^\ast(p;(\mathbb C^m,E),(M,D))\) denote the subset of all \(f\in\mathcal F(p;(\mathbb C^m,E),(M,D))\) that are analytically nondegenerate. The main result of the paper is the following theorem: If \(\text{rank} \varPsi=\dim M\) and \(d>(s+1)!\{(s+1)!-2\},\) then the number of mappings in \(\mathcal F^\ast(p;(\mathbb C^m,E),(M,D))\) is bounded by a constant depending only on \(D\). Remark. For divisors \(E_1\) and \(E_2\) on \(\mathbb C^m\) we write \(E_1\equiv E_2\pmod p\) if there exists a divisor \(E'\) on \(\mathbb C^m\) such that \(E_1-E_2=pE'\). in special case of \(p=0, E_1\equiv E_2\pmod 0\) iff \(E_1=E_2.\) Let \(E\) be a nonzeroeffective divisor on \(\mathbb C^m.\) The finiteness problem for meromorphic mappings under the condition on the preimages of divisors was first studied by H. Cartan and R. Nevanlinna and they obtained a finiteness theorem for meromorphic functions on the complex plane \(\mathbb C.\) The finiteness theorem of Cartan-Nevanlinna states that there exist at most two meromorphic functions on \(\mathbb C\) that have the same inverse images with multiplicities for distinct three values in \(\mathbb P_1(\mathbb C).\) H.~Fujimoto generalized the theorem of Cartan-Nevanlinna to the case of meromorphic mappings of \(\mathbb C^m\) into complex projective spaces \(\mathbb P_n(\mathbb C)\) by making use of Borel's identity. He proved the finiteness of families of linearly nondegenerate meromorphic mappings of \(\mathbb C^m\) into \(\mathbb P_n(\mathbb C)\) with the same inverse images for some hyperplanes. In his results, the number of hyperplanes in general position is essential and must be larger than a certain number depending on the dimension of the projective spaces. Furthermore, the finiteness theorem of Fujimoto has been extended to the case of meromorphic mappings into a projective algebraic manifold. The author mainly deals with the finiteness problem for meromorphic mappings \(f\) of \(\mathbb C^m\) into a compact complex manifold \(M\) and for a divisor \(D\) on \(M\).