Fractional Chern insulators in alternating twisted multilayer MoTe$_{2}$

We study strongly correlated many-body states in alternating twisted trilayer and tetralayer MoTe$_{2}$. By sliding the top layer with respect to others and applying a perpendicular electric field, a variety of band structures can be realized. In many cases, the topmost hole band has unity Chern number and its quantum geometric properties can be tuned to some extent. Exact diagonalizations suggest that fractional Chern insulators are stabilized in certain parameter regimes but not in some regimes even when the band is topological. This contrast is attributed primarily to different quantum geometries as quantified by the trace condition. Our results demonstrate that sliding can serve as a useful knob for probing many-body states in moir\'e systems.