Fibrati ampi e varietà speciali

Let X be a complex projective manifold of dimension n and let E be an ample vector bundle of rank r \leq n-1 on X having a section whose zero locus Z is a smooth submanifold of the expected dimension n-r. A well known philosophy coming from the study of hyperplane sections suggests that if Z is special enough, in a sense to be defined, then X itself has to be special. Recently, H. Maeda and I proved results aimed at classifying all possible pairs (X,E) as above under the assumption that Z is one of the special varieties arising in adjunction theory. Some of these results are outlined here. Moreover, several examples and some partial results concerned with the case r=n, escaping from the above set-up, are discussed as a matter of speculation related to recent research in the classification of ample vector bundles with low top Chern class