A general framework is described which associates geometrical structures to
any set of D finite-dimensional hermitian matrices Xa, a=1,...,D. This
framework generalizes and systematizes the well-known examples of fuzzy spaces,
and allows to extract the underlying classical space without requiring the
limit of large matrices or representation theory. The approach is based on the
previously introduced concept of quasi-coherent states. In particular, a
concept of quantum K\"ahler geometry arises naturally, which includes the
well-known quantized coadjoint orbits such as the fuzzy sphere SN2​ and
fuzzy CPNn​. A quantization map for quantum K\"ahler geometries is
established. Some examples of quantum geometries which are not K\"ahler are
identified, including the minimal fuzzy torus.Comment: 35 pages. V2: minor correction