Distribution des valeurs de transformations méromorphes et applications

• Autores: Tien-Cuong Dinh, Nessim Sibony
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 81, Nº 1, 2006, págs. 221-258
• Idioma: francés
• DOI: 10.4171/cmh/50
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• Resumen

  • 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 $\XX:=\prod_{n\geq 1} X_n$. 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 $\xx=(x_1,x_2,\ldots)$ and $\xx'=(x_1',x_2',\ldots)$ 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 {\it 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