  We formulate a relation between quantum-mechanical coherent states and
complex-differentiable structures on the classical phase space ${\cal C}$ of a
finite number of degrees of freedom. Locally-defined coherent states
parametrised by the points of ${\cal C}$ exist when there is an almost complex
structure on ${\cal C}$. When ${\cal C}$ admits a complex structure, such
coherent states are globally defined on ${\cal C}$. The picture of quantum
mechanics that emerges allows to implement duality transformations.
