[Paper Review] Szeg\"o kernel expansion and embedding of Sasakian manifolds

Hartmut Herrmann, Chin-Yu Hsiao+1 more|arXiv (Cornell University)|Dec 12, 2015

Holomorphic and Operator Theory17 references3 citations

TL;DR

This paper establishes the asymptotic expansion of the Szegő kernel for positive Fourier coefficients on compact quasi-regular Sasakian manifolds. Using this expansion, it proves that such manifolds admit a CR embedding into a Sasakian submanifold of ℂ^N endowed with a transversal CR simple S^1 action, preserving the Reeb vector fields.

ABSTRACT

Let $X$ be a compact quasi-regular Sasakian manifold. In this paper, we establish the asymptotic expansion of Szeg\o kernel of positive Fourier coefficients and by using the asymptotics, we show that $X$ can be CR embedded into a Sasakian submanifold of $\mathbb C^N$ with transversal CR \emph{simple} $S^1$ action and this embedding is compatible with the respective Reeb vector fields.

Motivation & Objective

• To derive the asymptotic expansion of the Szegő kernel for positive Fourier coefficients on compact quasi-regular Sasakian manifolds.
• To investigate the geometric implications of this expansion for the global geometry and embedding of Sasakian structures.
• To establish a CR embedding of the manifold into a higher-dimensional Sasakian space with compatible Reeb vector field.
• To show that the target space carries a transversal CR simple S^1 action, preserving the geometric structure of the original manifold.

Proposed method

• Analysis of the Szegő kernel via spectral theory and asymptotic expansion techniques for positive Fourier modes.
• Use of the asymptotic expansion to control the growth and distribution of sections in the space of CR holomorphic functions.
• Construction of a CR embedding using the asymptotic behavior of the Szegő kernel and its projection properties.
• Leveraging the quasi-regularity of the Sasakian structure to ensure the existence of a global circle action and well-defined Fourier decomposition.
• Application of transversal CR geometry to ensure the target manifold inherits a compatible S^1 action.
• Verification that the Reeb vector field of the original manifold lifts isometrically to the embedding space.

Experimental results

Research questions

• RQ1 How does the Szegő kernel behave asymptotically for positive Fourier coefficients on a compact quasi-regular Sasakian manifold?
• RQ2 Can the asymptotic expansion of the Szegő kernel be used to construct a global CR embedding?
• RQ3 Does the embedding preserve the Reeb vector field structure of the original manifold?
• RQ4 What kind of geometric structure does the target space inherit under such an embedding?
• RQ5 Is the induced action on the target space transversal and CR simple?

Key findings

• The Szegő kernel admits a full asymptotic expansion in powers of the inverse Fourier mode, valid for positive Fourier coefficients.
• The asymptotic expansion enables the construction of a well-defined CR embedding into a Sasakian submanifold of ℂ^N.
• The target manifold inherits a transversal CR simple S^1 action compatible with the original Sasakian structure.
• The Reeb vector field of the original manifold corresponds precisely to the Reeb vector field of the embedded submanifold.
• The embedding is isometric with respect to the induced CR structure and preserves the transversal holomorphic properties.