On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics

TL;DR

This paper develops a method to complexify Hamiltonian flows on compact real analytic symplectic manifolds, linking them to geodesics in the space of Kähler metrics and extending geometric quantization.

Contribution

It introduces a new approach using Lie series to analyze complexified Hamiltonian flows and their relation to Kähler structures, including non-analytic Hamiltonians.

Findings

Explicit formula for complex time evolution of Kähler potential.

Connection between complexified flows and geometric quantization.

Verification of geodesic families in Kähler structures.

Abstract

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.