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 $F_n:X\to X_n$ 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 $\mathbf{X}:={\prod }_{n\ge 1}{X}_n$. We give conditions which imply that

for $\sigma$-almost every $\mathbf{x}=(x_1,x_2,\dots )$ and ${\mathbf{x}}^{\prime }=(x_1^{\prime },x_2^{\prime },\dots )$ in $\mathbf{X}$. 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 $d{d}^c$-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 $f:X\to X$ is a meromorphic correspondence of large topological degree $d_t$, we show that $d_t^{-n}(f^n{)}^{\ast }{\omega }^k$ converges to a measure $\mu$, satisfying $f^{\ast }\mu =d_t\mu$. 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.