  We study Chern insulators from the point of view of K\"ahler geometry, i.e.
the geometry of smooth manifolds equipped with a compatible triple consisting
of a symplectic form, an integrable almost complex structure and a Riemannian
metric. The Fermi projector, i.e. the projector onto the occupied bands,
provides a map to a K\"ahler manifold. The quantum metric and Berry curvature
of the occupied bands are then related to the Riemannian metric and symplectic
form, respectively, on the target space of quantum states. We find that the
minimal volume of a parameter space with respect to the quantum metric is $\pi
|\mathcal{C}|$, where $\mathcal{C}$ is the first Chern number. We determine the
conditions under which the minimal volume is achieved both for the Brillouin
zone and the twist-angle space. The minimal volume of the Brillouin zone,
provided the quantum metric is everywhere non-degenerate, is achieved when the
latter is endowed with the structure of a K\"ahler manifold inherited from the
one of the space of quantum states. If the quantum volume of the twist-angle
torus is minimal, then both parameter spaces have the structure of a K\"ahler
manifold inherited from the space of quantum states. These conditions turn out
to be related to the stability of fractional Chern insulators. For two-band
systems, the volume of the Brillouin zone is naturally minimal provided the
Berry curvature is everywhere non-negative or nonpositive, and we additionally
show how the latter, which in this case is proportional to the quantum volume
form, necessarily has zeros due to topological constraints.
