  The fractional quantum anomalous Hall (FQAH) states or fractional Chern
insulator (FCI) states have been studied on two-dimensional (2D) flat lattices
with different boundary conditions. Here, we propose the geometry-dependent
FCI/FQAH states that interacting particles are bounded on 2D singular lattices
with arbitrary $n$-fold rotational symmetry. Based on the generalized Pauli
principle, we construct trial wave functions for the singular-lattice FCI/FQAH
states with the aid of an effective projection approach, and compare them with
the exact diagonalization results. High wave-function overlaps show that the
singular-lattice FCI/FQAH states are certainly related to the geometric factor
$\beta$. More interestingly, we observe some exotic degeneracy sequences of
edge excitations in these singular-lattice FCI/FQAH states, and provide an
explanation that two branches of edge excitations mix together.
