  The G3-property of a subvariety was introduced by Hironaka-Matsumura, and
plays an important role for deducing connectedness and extension results.
Unfortunately, it's a rather elusive notion, which is not always easy to
establish. Most of the existing work is concentrated on subvarieties of
homogeneous varieties.
  The first goal of this article is to show that mobility assumptions on the
subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a
certain partial positivity property, slightly stronger than G3, previously
introduced by the author. Second, we apply the result to prove that, in
numerous situations, the splitting of the normal bundle of a smooth
two-codimensional subvariety implies that it is a complete intersection.
