  We perform a theoretical study of the $\chi_{cJ} \to \phi K^* \bar K \to \phi
K\pi \bar K$ reaction taking into account the $K^* \bar K$ final state
interaction, which in the chiral unitary approach is responsible, together with
its coupled channels, for the formation of the low lying axial vector mesons,
in this case the $h_1(1380)$ given the selection of quantum numbers. Based on
this picture we can easily explain why in the $\chi_{c0}$ decay the $h_1(1380)$
resonance is not produced, and, in the case of $\chi_{c1}$ and $\chi_{c2}$
decay, why a dip in the $K^+ \pi^0 K^-$ mass distribution appears in the
1550-1600 MeV region, that in our picture comes from a destructive interference
between the tree level mechanism and the rescattering that generates the
$h_1(1380)$ state. Such a dip is not reproduced in pictures where the nominal
$h_1(1380)$ signal is added incoherently to a background, which provides
support to the picture where the resonance appears from rescattering of
vector-pseudoscalar components.
