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Equivariant Kodaira embedding of CR manifolds with circle action

Chin-Yu Hsiao, Xiaoshan Li, George Marinescu

Abstract

We consider a compact CR manifold with a transversal CR locally free circle action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.

Equivariant Kodaira embedding of CR manifolds with circle action

Abstract

We consider a compact CR manifold with a transversal CR locally free circle action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira embedding theorem.

Paper Structure

This paper contains 11 sections, 26 theorems, 256 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Preliminaries
  3. Some standard notations
  4. Definitions and notations from semi-classical analysis
  5. CR manifolds with circle action
  6. Expression of $T$ and $\overline\partial_b$ in BRT trivializations
  7. Szegő kernel asymptotics
  8. Small spectral gap of the Kohn Laplacian
  9. The operator $F_{k,\delta}$ on canonical coordinate patch
  10. Properties of the phase function
  11. Equivariant Kodaira embedding

Key Result

Theorem 1.1

Let $X$ be a compact CR manifold with a transversal CR locally free $S^1$-action and let $L$ be a positive rigid CR line bundle on $X$. Consider a point $p\in X$ and a canonical coordinate neighborhood $(D,x=(x_1,\ldots,x_{2n-1}))$ centered at $p=0$. Let $s$ be a local rigid CR frame of $L$ on $D$ a where $\varphi\in C^\infty( D\times D\times(-\delta,\delta))$ is a phase function such that for som

Theorems & Definitions (55)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 45 more