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

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.

final version

References in corpus (6)

• Lectures on Stability and Constant Scalar Curvature
• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan
• The exponential map of the complexification of {\em Ham} in the real-analytic case
• Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds
• Coherent state transforms and the Mackey-Stone-Von Neumann theorem

Cited by in corpus (2)

• Complex structures adapted to magnetic flows
• Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds