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 Fn ⁣:XXnF_n\colon X\rightarrow X_n be a sequence of MT. Let σn\sigma_n be an appropriate probability measure on XnX_n and σ\sigma the product measure of σn\sigma_n, on \XX:=n1Xn\XX:=\prod_{n\geq 1} X_n. We give conditions which imply that

1d(Fn)[(Fn)(δxn)(Fn)(δxn)]0\frac{1}{d(F_n)}\big[(F_n)^*(\delta_{x_n})-(F_n)^*(\delta_{x_n'})\big]\rightarrow 0

for σ\sigma-almost every \xx=(x1,x2,)\xx=(x_1,x_2,\ldots) and \xx=(x1,x2,)\xx'=(x_1',x_2',\ldots) in X\boldsymbol{X}. Here δxn\delta_{x_n} is the Dirac mass at xnx_n and d(Fn)d(F_n) the intermediate degree of maximal order of FnF_n. We introduce a calculus on MT: intermediate degrees of composition and of product of MT. Using this formalism and what we call {\it the ddcdd^c-method}, we obtain results on the distribution of common zeros, for random ll holomorphic sections of high powers LnL^n of a positive holomorphic line bundle LL over a projective manifold. We also construct the equilibrium measure for random iteration of correspondences. In particular, when f ⁣:XXf\colon X\rightarrow X is a meromorphic correspondence of large topological degree dtd_t, we show that dtn(fn)ωkd_t^{-n}(f^n)^*\omega^k converges to a measure μ\mu, satisfying fμ=dtμf^*\mu=d_t\mu. Moreover, quasi-psh functions are μ\mu-integrable. Every projective manifold admits such correspondences. When ff is a meromorphic map, μ\mu 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