  Within the framework of perturbative QCD approach, we study the charmless
two-body decays $B\to a_1(1260)K^*, b_1(1235)K^*$. Using the decays constants
and the light-cone distribution amplitudes for these mesons derived from the
QCD sum rule method, we find the following results: (a) Our predictions for the
branching ratios are consistent well with the QCDF results within errors, but
much larger than the naive factorization approach calculation values. (b) We
predict that the anomalous polarizations occurring in the decays $B\to \phi
K^*, \rho K^*$ also happen in the decays $B\to a_1K^*$, while do not happen in
the decays $B\to b_1K^*$. Here the contributions from the annihilation diagrams
play an important role to explain the lager transverse polarizations in the
decays $B\to a_1K^*$, while they are not sensitive to the polarizations in
decays $B\to b_1K^*$. (c) Our predictions for the direct CP-asymmetries agree
well with the QCDF results within errors. The decays $\bar B^0\to
b^{+}_1K^{*-}, B^-\to b^{0}_1K^{*-}$ have larger direct CP-asymmetries, which
could be measured by the present LHCb experiments.
