  Our first goal in this note is to explain that a weak form of Perrin-Riou's
conjecture on the non-triviality of Beilinson-Kato classes follows as an easy
consequence of the Iwasawa main conjectures, and deduce its refined versions in
the supersingular case from this fact and a variety of Gross-Zagier formulae.
  Our second goal is to set up a conceptual framework in the context of
$\Lambda$-adic Kolyvagin systems to treat analogues of Perrin-Riou's
conjectures for higher motives of higher rank. We apply this general discussion
in order to establish a link between Heegner points on a general class of CM
abelian varieties and the (conjectural) Coleman-Rubin-Stark elements we
introduce here. This is a higher dimensional version of Rubin's results on
rational points on CM elliptic curves.
