  The extended $t-J$ model is theoretically studied, in the context of hole
underdoped cuprates. Based on results obtained by recent numerical studies, we
identify the mean field state having both the antiferromagnetic and staggered
flux resonating valence bond orders. The random-phase approximation is employed
to analyze all the possible collective modes in this mean field state. In the
static (Bardeen Cooper Schrieffer) limit justified in the weak coupling regime,
we obtain the effective superconducting interaction between the doped holes at
the small pockets located around $\bm{k}= (\pm \pi/2, \pm \pi/2)$. In contrast
to the spin-bag theory, which takes into acccount only the antiferromagnetic
order, this effective force is pair breaking for the pairing without the nodes
in each of the small hole pocket, and is canceled out to be very small for the
$d_{x^2-y^2}$ pairing with nodes which is realized in the real cuprates.
Therefore we conclude that no superconducting instability can occur when only
the magnetic mechanism is considered. The relations of our work with other
approaches are also discussed.
