  We study twisted bilayer MoTe$_2$ systems at fractional fillings of the
lowest hole band under applied out-of-plane displacement fields. By employing
exact diagonalization in finite-size systems, we systematically map out the
ground state quantum phase diagram for two filling fractions, $\nu=1/3$ and
$2/3$, and provide a detailed characterization of each phase. We identify the
phase transition between a fractional Chern insulator (FCI) and a
layer-polarized charge density wave (CDW) at a filling fraction of $\nu=1/3$,
denoted as CDW-$1$. Additionally, we demonstrate that the competition between
the displacement field and twist angle leads to another phase transition from a
layer-polarized CDW-$1$ to a layer-hybridized CDW-$2$, identified as a
first-order phase transition. Furthermore, at $\nu=2/3$ filling of the lowest
hole band, we observe that the FCI remains stable against the displacement
field until it approaches proximity to a transition in single-particle band
topology at a smaller twist angle.
