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Uniqueness problem for meromorphic mappings with truncated multiplicities and few targets. (English) Zbl 1126.32013

The paper under review deals with uniqueness problem for meromorphic mappings of \(\mathbb C^m\) into \(\mathbb P^n(\mathbb C)\) with truncated multiplicities. Let \(n,x,y\) and \(p\) be non-negative integers such that \(2\leq p\leq n\), \(1\leq y\leq 2n\), \(0\leq x< \min\{2n-y+1,(p-1)y/ (n+1+y)\}\). Let \(k\) be a positive integer or \(+\infty\) with \((2n(n+1+y) (3n+p-x))/((p-1)y-x(n+1+y))\leq k\leq+\infty\). Let \(f,g:\mathbb C^m\to \mathbb P^n(\mathbb C)\) be linearly nondegenerate meromorphic mappings and \(\{H_j\}^q_{j=1}\) hyperplanes in general position in \(\mathbb P^n(\mathbb C)\). Let \(\nu(f,H_j)(z)\) be the intersection multiplicity of the image of \(f\) and \(H_j\) at \(f(z)\) and thus \(\nu(f,H_j)\) is a \(\mathbb Z\)-valued function on \(\mathbb C^m\). For \(j=1,\dots,q\), we set \(E^j_f=\{z\in\mathbb C^m: 0\leq\nu (f,H_j)(z)\leq k\}\) and \(*E^j_f= \{z\in\mathbb C^m:0<\nu(f,H_j)(z)\leq k\}\). We also define \(E^j_g\) and \(*E^j_g\) as in the above. Assume that \(\dim (*E^i_f\cap *E^j_f)\leq m-2\) and \(\dim(*E^i_g\cap *E^j_g)\leq m-2\) 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 \(q\leq 3n+1-x\). If \(\min\{\nu(f,H_j),1\}=\min\{\nu(g,H_j),1\}\) on \(E^j_f \cap E^j_q\) for \(j=n+2+y,\dots,q\), and if \(\min\{\nu(f,H_j), p\}=\min\{\nu(g, H_j),p\}\) on \(E^j_f\cap E^j_g\) for \(j=1,\dots,n+1+ y\), then \(f=g\).
The authors also consider the moving target case and give results similar to the above.

MSC:

32H04 Meromorphic mappings in several complex variables
32H30 Value distribution theory in higher dimensions
32H25 Picard-type theorems and generalizations for several complex variables
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References:

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