  In the case of a compact real analytic symplectic manifold M we describe an
approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and
corresponding geodesics on the space of Kahler metrics. In this approach,
motivated by recent work on quantization, the complexified Hamiltonian flows
act, through the Grobner theory of Lie series, on the sheaf of complex valued
real analytic functions, changing the sheaves of holomorphic functions. This
defines an action on the space of (equivalent) complex structures on M and also
a direct action on M. This description is related to the approach of [BLU]
where one has an action on a complexification M_C of M followed by projection
to M. Our approach allows for the study of some Hamiltonian functions which are
not real analytic. It also leads naturally to the consideration of continuous
degenerations of diffeomorphisms and of Kahler structures of M. Hence, one can
link continuously (geometric quantization) real, and more general non-Kahler,
polarizations with Kahler polarizations. This corresponds to the extension of
the geodesics to the boundary of the space of Kahler metrics. Three
illustrative examples are considered. We find an explicit formula for the
complex time evolution of the Kahler potential under the flow. For integral
symplectic forms, this formula corresponds to the complexification of the
prequantization of Hamiltonian symplectomorphisms. We verify that certain
families of Kahler structures, which have been studied in geometric
quantization, are geodesic families.
