Some uniqueness theorems with truncated multiplicities of meromorphic mappings (Q950683)

The paper under review deals with uniqueness problems for meromorphic mappings of \(\mathbb{C}^n\) into \(\mathbb{P}^N(\mathbb{C})\) with truncated multiplicities. Let \(f,g:\mathbb{C}^n \to\mathbb{P}^N(\mathbb{C})\) be linearly non-degenerate meromorphic mappings and \(\{H_j\}^q_{j=1}\) hyperplanes in general position in \(\mathbb{P}^N(\mathbb{C})\). Let \(x,y\) and \(p\) be non-negative integers such that \(2\leq p\leq N\), \(1\leq y\leq 2N\) and \(0\leq x<\min\{2N-y+1,(p-1)y/(N+1+y)\}\). Let \(k\) be a positive integer or \(+\infty\). Let \(\nu(f,H_j)(z)\) be the intersection multiplicity of the image of \(f\) and \(H_j\) at \(f(z)\). For \(j=1,\dots,q\), we set \(E_f^j=\{z\in \mathbb{C}^n:0\leq\nu (f,H_j)(z)\leq k\}\) and \(^*E^j_f=\{z\in\mathbb{C}^n:0<\nu (f,H_j)(z)\leq k\}\). We also define \(E^j_g\) and \(^*E^j_g\) analogously. Assume that \(\dim(^*E^i_f\cap{^*E^j_f})\leq n-2\) and \(\dim (^*E^i_g\cap{^*E^j_g})\leq n-2\) for all \(i\neq j\). We also assume \(f=g\) on \(\bigcup^q_{j=1}(^*E^j_f\cap{^*E^j_g})\). Then the main result in this paper can be stated as follows: Suppose that \(\min\{\nu(f,H_j),1\}= \min\{\nu(g,H_j),1\}\) on \(E^j_f\cap E^j_g\) for \(j=N+2+y,\dots,q\) and \(\min\{\nu(f,H_j), p\}=\min\{\nu(g,H_j),p\}\) on \(E^j_f\cap E^j_g\) for \(j=1,\dots,N+1+y\). If \(q\geq 3N+1-x\), then \(f\equiv g\) for \(k\geq (2N(N+1+y)(3N+p/2-x))/((p-1)y-x(N+1+y))\). This gives us an improvement of a theorem obtained by \textit{G. Dethloff} and \textit{T. Van Tan} [Ann. Fac. Sci. Toulouse, Math. (6) 15, No.~2, 217--242 (2006; Zbl 1126.32013)].