Multiplicity of holomorphic functions
Let \(f:P^2\to P^2\) be a rational map of maximal rank and of degree \(d\). Denote by \(I(f)\) the (finite) set of points of indeterminancy of \(f\). The map \(f\) is said to be algebraically stable if \(f^{-n}(I(f))\) is finite for any \(n\geq 0\). Under this condition, it is possible to define an invariant positive closed (1,1)-current \(T(f)=\lim_{n \to\infty} T_n(f)\), where \(T_n(f)= d^{-n}(f^n)^*w\) and \(w\) is the Fubini-Study form. Let \(\nu(T(f),z)\) denote the Lelong number of \(T(f)\) at \(z\) and \(I(f^\infty)= \cup_{n\geq 0}f^{-n} (I(f^n))\). Theorem 1. \(\nu(T(f),z)>0\) if and only if \(z\in I(f^\infty)\). Theorem 3. There exists a function \(\mu_\infty: I(f^\infty) \to\mathbb{R}^*_+\) to the positive reals of total mass \(\sum_{z\in I(f^\infty)}\mu_\infty (z)=1\) such that \[ \lim_{n\to \infty} T_n(f)\wedge T_n(f)=\sum_{z\in I(f^\infty)} \mu_\infty(z) \delta_z. \]
scientific article; zbMATH DE number 1423135