Pluripolar set

Abstract: The main result in the article isTheorem. Let be a closed set such that and is a pseudoconvex domain. If for almost every complex line passing through 0 the intersection is polar in , then is a pluripolar set in .This theorem is then applied to the analysis of sets of singularities of holomorphic functions which are rapidly approximated by rational functions.Bibliography: 21 titles.

Abstract: We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kahler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.