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Our first goal in this note is to explain that a weak form of Perrin-Riou'sconjecture on the non-triviality of Beilinson-Kato classes follows as an easyconsequence of the Iwasawa main conjectures, and deduce its refined versions inthe 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'sconjectures for higher motives of higher rank. We apply this general discussionin order to establish a link between Heegner points on a general class of CMabelian varieties and the (conjectural) Coleman-Rubin-Stark elements weintroduce here. This is a higher dimensional version of Rubin's results onrational points on CM elliptic curves.

Beilinson-Kato and Beilinson-Flach elements, Coleman-Rubin-Stark classes, Heegner points and the Perrin-Riou Conjecture