Equivariant K\"ahlerian extensions of contact manifolds

TL;DR

This paper constructs a complexified extension of contact manifolds where the contact form extends to a K"ahler form, and demonstrates an equivariant reduction process linking contact and K"ahlerian reductions.

Contribution

It introduces a method to complexify contact manifolds with an extended K"ahler form, preserving equivariance under Lie group actions.

Findings

Construction of complexification with K"ahlerian exterior derivative

Equivariant realization for extendable Lie group actions

K"ahlerian reduction as complexification of contact reduction

Abstract

For contact manifolds $(M, \eta)$ a complexification $M^c$ is constructed to which the contact form $\eta$ extends such that the exterior derivative of the extended form is K\"ahlerian. In the case of a proper action of an extendable Lie group this construction is realized in an equivariant way. In a simultaneous stratification of $M$ and $M^c$ according to the istropy type, it is shown that the K\"ahlerian reduction of the complexification can be seen as the complexification of the contact reduction.