Value distribution of meromorphic transforms and applications

Abstract

A meromorphic transform between complex manifolds is a surjective mutivalued map with an analytic graph. Let $F_n$ be a sequence of meromorphic transforms from a compact Kahler manifold X into compact Kahler manifolds X_n. We give conditions which imply that the behavior of the sequence of preimages $F_n^{-1}(x_n)$ of x_n does not depend on the generic sequence of points (x_1,x_2,....). Using this formalism, we obtain sharp results on the limit distribution of common zeros, of $l$ random holomorphic sections of high powers $L^n$ of a positive holomorphic line bundle L over a projective manifold X. We consider also the equidistribution problem for random iteration of correspondences. If f is a meromorphic self correspondence of a compact Kahler manifold X, under a hypothesis on the dynamical degrees, we construct an $f^*$-invariant probability measure $μ$ such that quasi-p.s.h. functions are $μ$-integrable. Every projective manifold admits such correspondences. When f is a meromorphic map, the measure $μ$ is exponentially mixing. We give some analogous results for random iterations of correspondences. We also consider the problem of equidistribution of preimages of subvarieties for a correspondences and more precisely for polynomial automorphisms.