Multiplicity of holomorphic functions. (English) Zbl 0948.32020

Favre, Charles

Math. Ann. 316, No. 2, 355-378 (2000).

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 \[ \lim_{n\to \infty} T_n(f)\wedge T_n(f)=\sum_{z\in I(f^\infty)} \mu_\infty(z) \delta_z. \]

Reviewer: S.M.Ivashkovich (Villeneuve d’Ascq)

MSC:

32H50

Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables

Keywords:

iteration; rational map; Lelong number