  Looking for a quantum-mechanical implementation of duality, we formulate a
relation between coherent states and complex-differentiable structures on
classical phase space ${\cal C}$. A necessary and sufficient condition for the
existence of locally-defined coherent states is the existence of an almost
complex structure on ${\cal C}$. A necessary and sufficient condition for
globally-defined coherent states is a complex structure on ${\cal C}$.
  The picture of quantum mechanics that emerges is conceptually close to that
of a geometric manifold covered by local coordinate charts. Instead of the
latter, quantum mechanics has local coherent states. A change of coordinates on
${\cal C}$ may or may not be holomorphic. Correspondingly, a transformation
between quantum-mechanical states may or may not preserve coherence. Those that
do not preserve coherence are duality transformations. A duality appears as the
possibility of giving two or more, apparently different, descriptions of the
same quantum-mechanical phenomenon. Coherence becomes a local property on
classical phase space. Observers on ${\cal C}$ not connected by means of a
holomorphic change of coordinates need not, and in general will not, agree on
what is a semiclassical effect vs. what is a strong quantum effect
