  Iwasawa theory of Heegner points on abelian varieties of GL_2 type has been
studied by, among others, Mazur, Perrin-Riou, Bertolini and Howard. The purpose
of this paper, the first in a series of two, is to describe extensions of some
of their results in which abelian varieties are replaced by the Galois
cohomology of Deligne's p-adic representation attached to a modular form f of
even weight >2. In this more general setting, the role of Heegner points is
played by higher-dimensional Heegner cycles in the sense of Nekov\'a\v{r}. In
particular, we prove that the Pontryagin dual of a certain Bloch-Kato Selmer
group associated with f has rank 1 over a suitable anticyclotomic Iwasawa
algebra.
