Value distribution of meromorphic transforms and applications

الأصل · EN

A meromorphic transform between complex manifolds is a surjective mutivalued map with an analytic graph. Let Fₙ be a sequence of meromorphic transforms from a compact Kahler manifold X into compact Kahler manifolds Xₙ. We give conditions which imply that the behavior of the sequence of preimages Fₙ⁻¹(xₙ) of xₙ does not depend on the generic sequence of points (x₁,x₂,....). Using this formalism, we obtain sharp results on the limit distribution of common zeros, of l random holomorphic sections of high powers Lⁿ 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.