  Let $p>3$ be a prime. In this paper we develop a new kind of anticyclotomic
local $\pm$-Iwasawa theory at $p$ for Hecke characters of quadratic imaginary
fields which is valid for all ramification types of $p$ (split, inert and
ramified). As an application we deduce the converse of Gross-Zagier-Kolyvagin
theorem for these CM forms, which states that Selmer rank one implies analytic
rank one. To carry out the Iwasawa theory argument we employ a recent
construction of a new type of $p$-adic $L$-function by Andreatta-Iovita, and
generalized by Yangyu Fan to Shimura curves in the Appendix, and a ``virtual
Heenger family'' made via a limiting procedure from a Heegner family along
Coleman-Mazur eigencurve constructed by Jetchev-Loeffler-Zerbes.
