Szegő kernel expansion and equivariant embedding of CR manifolds with circle action

Abstract

We consider a compact connected CR manifold with a transversal CR locally free R-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 and we establish R-equivariant Kodaira embedding theorem for CR manifolds. Using similar methods we also establish an analytic proof of an R-equivariant Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In particular, we obtain equivariant embedding theorems for irregular Sasakian manifolds. As applications of our results, we obtain Torus equivariant Kodaira and Boutet de Monvel embedding theorems for CR manifolds and Torus equivariant Kodaira embedding theorem for complex manifolds.

Asymptotics of torus equivariant Szegö kernel on a compact CR manifold

“…Suppose now X is irregular and T be the fundamental vector field of the R-action. Take a R-invariant L 2 -inner product on X and consider the weak maximal extension of T on L 2 functions, then in [11] it was shown that T is a self-adjoint operator, and the spectrum of T, denoted by Spec(T), is a countable subset in R. Moreover, all the spectrum of T are eigenvalues. On irregular Sasakian manifolds, it is important to understand the space H 0 b,α (X) := {u ∈ C ∞ (X) : ∂ b u = 0, Tu = iαu}, where ∂ b denotes the tangential Cauchy-Riemann operator on X.…”

“…Proof. The argument is almost the same as the case [5, Section 3] when only circle action is involved, and for the modification to torus action we refer to the proof in [11,Section 4].…”

Torus Equivariant Szegő Kernel Asymptotics on Strongly Pseudoconvex CR Manifolds

“…Assume that X admits a compact connected Lie group action G. The study of G-equivariant CR functions and Szegő kernel is closely related to some problems in CR, complex geometry, Mathematical physics and geometric quantization theory. For example, for a compact irregular Sasakian manifold X, it was shown in [4] that X admits a torus action T d and the study of torus-equivariant CR functions and Szegő kernel is important in Sasaki geometry. In this work, we consider a compact strongly pseudoconvex CR manifold (X, T 1,0 X) of dimension 2n + 1 and assume that X admits a torus action T d = (e iθ 1 , .…”

“…Let S k,τ (x, y) ∈ D ′ (X × X) be the distribution kernel of S k,τ . For every λ ∈ R + , it was shown in Lemma 4.6 in [4] that the…”

On a Holomorphic Family of Stein Manifolds with Strongly Pseudoconvex Boundaries

“…On the other hand, if a compact three dimensional strongly pseudoconvex CR manifold admits a transversal CR S 1 -action, it was shown by Lempert [Le92], Epstein [Ep92] and recently in [HM14,HHL15] by using the Szegő kernel, that such CR manifolds can always be CR embedded into a complex Euclidean space.…”