  A recent surge of research in many-body quantum entanglement has uncovered
intriguing properties of quantum many-body systems. A prime example is the
modular commutator, which can extract a topological invariant from a single
wave function. Here, we unveil novel geometric properties of many-body
entanglement via a modular commutator of two-dimensional gapped quantum
many-body systems. We obtain the geometric additivity of a modular commutator,
indicating that modular commutator for a multipartite system may be an integer
multiple of the one for tripartite systems. Using our additivity formula, we
also derive a curious identity for the modular commutators involving
disconnected intervals in a certain class of conformal field theories. We
further illustrate this geometric additivity for both bulk and edge subsystems
using numerical calculations of the Haldane and $\pi$-flux models.
