Nearly Kahlerian Embeddings of Symplectic Manifolds

TL;DR

This paper introduces a new method for embedding integral symplectic manifolds into complex projective space using coherent states, creating an almost-complex analogue of the Kodaira embedding with approximate pseudo-holomorphic properties.

Contribution

It constructs a family of embeddings from sections of tensor powers of a hermitian line bundle, extending the classical Kodaira embedding to an almost-complex setting.

Findings

The embeddings are approximately pseudo-holomorphic.

The construction generalizes the Kodaira embedding to symplectic manifolds.

The maps are shown to be genuine embeddings.

Abstract

Given an integral symplectic manifold, we construct a family of "coherent state" maps into complex projective space. The maps are built from sections of the tensor powers of a hermitian line bundle whose curvature is a multiple of the symplectic form. We show that this family is an almost-complex version of the Kodiara embedding. That is, the maps are embeddings which are approximately pseudo-holomorphic.