Projections in several complex variables

Hsiao, Chin-Yu

doi:10.24033/msmf.435

Cited by 28 publications

(80 citation statements)

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“…In this case the result of [44] applies and we can describe the Szegő kernel Π (q) (x, y). On the negative side, Burns [19] showed that the closed range property fails in the case of the non-embeddable exemples of Grauert, Andreotti-Siu and Rossi [36,1,62].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Theorems 3.1, Theorem 3.2 and Theorem 3.4 are proved in chapter 6, chapter 7 and chapter 8 of part I in [44]. In [44] the existence of the microlocal Hodge decomposition is stated for compact CR manifolds, but the construction and arguments used are essentially local.…”

Section: Microlocal Hodge Decomposition Theorems For (Q) Bmentioning

confidence: 99%

“…In Section 4, we first study the microlocal behahaviour of the spectral function and by using the microlocal Hodge decomposition of the Kohn Laplacian established in [44], we prove Theorem 1.5. Furthermore, by using Theorem 1.5 and some standard technique in functional analysis, we prove Theorem 1.7.…”

Section: Theorem 17mentioning

confidence: 99%

“…In [44] the existence of the microlocal Hodge decomposition is stated for compact CR manifolds, but the construction and arguments used are essentially local.…”

Section: Microlocal Hodge Decomposition Theorems For (Q) Bmentioning

confidence: 99%

“…In Section 3, we review some results in [44] about the existence of a microlocal Hodge decomposition of the Kohn Laplacian on an open set of a CR manifold where the Levi form is non-degenerate.…”

Section: Theorem 17mentioning

confidence: 99%

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On the singularities of the Szegő projections on lower energy forms

Hsiao

Marinescu

2017

J. Differential Geom.

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ABSTRACT. Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n − 1, n 2. Let (q) b be the Gaffney extension of Kohn Laplacian for (0, q)-forms. We show that the spectral function of (q) b admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if X is compact and the Levi form is nondegenerate of constant signature on X, then the spectrum ofin ]0, ∞[ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szegő kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szegő kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR S 1 actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR S 1 action can be CR embedded into C N , for some N ∈ N.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Microlocal Hodge Decomposition Theorems For (Q) Bmentioning

confidence: 99%

Section: Theorem 17mentioning

confidence: 99%

“…In [44] the existence of the microlocal Hodge decomposition is stated for compact CR manifolds, but the construction and arguments used are essentially local.…”

Section: Microlocal Hodge Decomposition Theorems For (Q) Bmentioning

confidence: 99%

Section: Theorem 17mentioning

confidence: 99%

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On the singularities of the Szegő projections on lower energy forms

Hsiao

Marinescu

2017

J. Differential Geom.

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Bergman Kernel Asymptotics and a Pure Analytic Proof of the Kodaira Embedding Theorem

Hsiao

2015

Springer Proceedings in Mathematics &Amp; Statistics

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Quotient of Bergman kernels on punctured Riemann surfaces

Auvray

Marinescu

2022

Math. Z.

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In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures, and a holomorphic line bundle that polarizes the metric. We introduce a new method to compare the Bergman kernels of high tensor powers of the line bundle and of the Poincaré model near the singularity and show that their quotient tends to one uniformly on a neighborhood of the singularity up to arbitrary negative powers of the tensor power.

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Projections in several complex variables

Cited by 28 publications

References 0 publications

On the singularities of the Szegő projections on lower energy forms

On the singularities of the Szegő projections on lower energy forms

Bergman Kernel Asymptotics and a Pure Analytic Proof of the Kodaira Embedding Theorem

Quotient of Bergman kernels on punctured Riemann surfaces

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