  The quantum metric of single-particle wave functions in topological flatbands
plays a crucial role in determining the stability of fractional Chern
insulating (FCI) states. Here, we unravel that the quantum metric causes the
many-body Chern number of the FCI states to deviate sharply from the expected
value associated with partial filling of the single-particle topological
flatband. Furthermore, the variation of the quantum metric in momentum space
induces band dispersion through interactions, affecting the stability of the
FCI states. This causes a reentrant transition into the Fermi liquid from the
FCI phase as the interaction strength increases.
