  A series of recent experimental works on twisted MoTe$_2$ homobilayers have
unveiled an abundance of exotic states in this system. Valley-polarized quantum
anomalous Hall states have been identified at hole doping of $\nu = -1$, and
the fractional quantum anomalous Hall effect is observed at $\nu = -2/3$ and
$\nu = -3/5$. In this work, we investigate the electronic properties of
AA-stacked twisted bilayer MoTe$_2$ at $\nu=-2$ by $k$-space Hartree-Fock
calculations. We find that the phase diagram is qualitatively similar to the
phase diagram of a Kane-Mele-Hubbard with staggered onsite potential. A
noteworthy phase within the diagram is the antiferromagnetic Chern insulator,
stabilized by the external electric field. We attribute the existence of this
Chern insulator to an antiferromagnetic instability at a topological phase
transition between the quantum spin hall phase and a band insulator phase. We
highlight that the antiferromagnetic Chern insulator phase is most evident at a
twist angle of approximately $4^\circ$. Our research proposes the potential of
realizing a Chern insulator beyond $\nu=-1$, and contributes fresh perspectives
on the interplay between band topology and electron-electron correlations in
moir\'e superlattices.
