Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Kevin Fritsch,Peter Heinzner

doi:10.1007/s00229-021-01291-w

Abstract

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal G rtimes S^1-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

Highlights

An important and much studied question in CR geometry is whether an abstract CR manifold can be realized, locally or even globally, as a CR submanifold of Cn, see for example [1,4] or [15]
There have been several works on the topic of CR manifolds with transversal group actions [2], Lempert proved an embedding result for the otherwise difficult 3-dimensional case assuming the existence of a transversal CR R-action [16]
There have been more recent results for CR manifolds with transversal S1action by Herrmann et al [12] and an equivariant Kodaira embedding theorem by Hsiao et al [13]

Summary

Introduction

An important and much studied question in CR geometry is whether an abstract CR manifold can be realized, locally or even globally, as a CR submanifold of Cn, see for example [1,4] or [15]. In this paper we will consider X to be a CR manifold with proper, transversal action of a group G such that G is a subgroup of its universal complexification GC. Let X be a CR manifold with proper, transversal, CR action of a Lie group G. There exists a universal equivariant extension for X We use this to show that the quotient space X/G carries the structure of a complex space such that the sheaf of holomorphic functions on X/G is given by the sheaf of G-invariant CR functions on X (Theorem 1.8). Using the quotient result and methods from complex geometry, we prove the following equivariant embedding theorem. Let X be a compact CR manifold with a transversal CR action of a compact Lie group K. We will show that the above embedding can be chosen to be K equivariant

Quotients

Pseudoconvexity

Equivariant embeddings

Line bundles