Distribution des valeurs de transformations méromorphes et applications

  • Tien-Cuong Dinh

    Institut Mathématique de Jussieu, Paris, France
  • Nessim Sibony

    Université Paris-Sud, Orsay, France


A meromorphic transform (MT) between compact Kähler manifolds is a surjective multivalued map with an analytic graph. Let be a sequence of MT. Let be an appropriate probability measure on and the product measure of , on \( \XX:=\prod_{n\geq 1} X_n \). We give conditions which imply that

for -almost every \( \xx=(x_1,x_2,\ldots) \) and \( \xx'=(x_1',x_2',\ldots) \) in . Here is the Dirac mass at and the intermediate degree of maximal order of . We introduce a calculus on MT: intermediate degrees of composition and of product of MT. Using this formalism and what we call {\it the -method}, we obtain results on the distribution of common zeros, for random holomorphic sections of high powers of a positive holomorphic line bundle over a projective manifold. We also construct the equilibrium measure for random iteration of correspondences. In particular, when is a meromorphic correspondence of large topological degree , we show that converges to a measure , satisfying . Moreover, quasi-psh functions are -integrable. Every projective manifold admits such correspondences. When is a meromorphic map, is exponentially mixing with a precise speed depending on the regularity of the observables.

Cite this article

Tien-Cuong Dinh, Nessim Sibony, Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv. 81 (2006), no. 1, pp. 221–258

DOI 10.4171/CMH/50