On Complexified Analytic Hamiltonian Flows and Geodesics on the Space of Kähler Metrics

Abstract

For a compact real analytic symplectic manifold |$(M,\omega ),$| we describe an approach to the complexification of Hamiltonian flows [5, 30, 33] and corresponding geodesics on the space of Kähler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Gröbner 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 [3] where one has an action on a complexification |$M_{\mathbb {C}}$| 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 Kähler structures of |$M$|⁠. Hence, one can link continuously (geometric quantization) real, and more general non-Kähler, polarizations with Kähler polarizations. This corresponds to the extension of the geodesics to the boundary of the space of Kähler metrics. Three illustrative examples are considered. We find an explicit formula for the complex time evolution of the Kähler 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 Kähler structures, which have been studied in geometric quantization, are geodesic families.