  Let $(M, \omega, J)$ be a K\"ahler manifold and K its group of hamiltonian
symplectomorphisms. The complexification of K introduced by Donadson is not a
group, only a "formal Lie group". However it still makes sense to talk about
the exponential map in the complexification. In this note we show how to
construct geometrically the exponential map (for small time), in case the
initial data are real-analytic. The construction is motivated by, but does not
use, semiclassical analysis.
