  Despite more than two decades of intensive investigations, the true nature of
high temperature (high-$T_c$) superconductivity observed in the cuprates
remains elusive to the researchers. In particular, in the so-called
`underdoped' region, the overall behavior of superconductivity deviates
$qualitatively$ from the standard theoretical description pioneered by Bardeen,
Cooper and Schrieffer (BCS). Recently, the importance of phase fluctuation of
the superconducting order parameter has gained significant support from various
experiments. However, the microscopic mechanism responsible for the
surprisingly soft phase remains one of the most important unsolved puzzles.
Here, opposite to the standard BCS starting point, we propose a simple,
solvable low-energy model in the strong coupling limit, which maps the
superconductivity literally into a well-understood physics of superfluid in a
special dilute bosonic system of local pairs of doped holes. In the
prototypical material (La$_{1-\delta}$Sr$_\delta$)$_2$CuO$_4$, without use of
any free parameter, a $d$-wave superconductivity is obtained for doping above
$\sim 5.2\%$, below which unexpected incoherent $p$-wave pairs dominate.
Throughout the whole underdoped region, very soft phases are found to originate
from enormous mass enhancement of the pairs. Furthermore, a striking mass
divergence is predicted that dictates the occurrence of the observed quantum
critical point. Our model produces properties of the superfluid in good
agreement with the experiments, and provides new insights into several current
puzzles. Owing to its simplicity, this model offers a paradigm of great value
in answering the long-standing challenges in underdoped cuprates.
