Distribution des valeurs de transformations méromorphes et applications

Abstract

A meromorphic transform (MT) between compact Kähler manifolds is a surjective multivalued map with an analytic graph. Let F_n\colon X\rightarrow X_n be a sequence of MT. Let \sigma_n be an appropriate probability measure on X_n and \sigma the product measure of \sigma_n , on \boldsymbol X:=\prod_{n\geq 1} X_n . We give conditions which imply that \frac{1}{d(F_n)}\big[(F_n)^*(\delta_{x_n})-(F_n)^*(\delta_{x_n'})\big]\rightarrow 0 for \sigma -almost every \boldsymbol x=(x_1,x_2,\ldots) and \boldsymbol x'=(x_1',x_2',\ldots) in \boldsymbol{X} . Here \delta_{x_n} is the Dirac mass at x_n and d(F_n) the intermediate degree of maximal order of F_n . We introduce a calculus on MT: intermediate degrees of composition and of product of MT. Using this formalism and what we call the dd^c -method , we obtain results on the distribution of common zeros, for random l holomorphic sections of high powers L^n of a positive holomorphic line bundle L over a projective manifold. We also construct the equilibrium measure for random iteration of correspondences. In particular, when f\colon X\rightarrow X is a meromorphic correspondence of large topological degree d_t , we show that d_t^{-n}(f^n)^*\omega^k converges to a measure \mu , satisfying f^*\mu=d_t\mu . Moreover, quasi-psh functions are \mu -integrable. Every projective manifold admits such correspondences. When f is a meromorphic map, \mu is exponentially mixing with a precise speed depending on the regularity of the observables.