An Isometric Imbedding Theorem for Holomorphic Bundles

Catlin, David W.; D’Angelo, John P.

doi:10.4310/mrl.1999.v6.n1.a4

Cited by 23 publications

(50 citation statements)

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“…In the language of this section, we can phrase the following beautiful result of Catlin and D'Angelo [CD2] as follows.…”

Section: Remarkmentioning

confidence: 99%

“…The class of SGCS metrics on E will be denoted P S 2 (X, E). Definition 2.10 is due essentially to Catlin and D'Angelo [CD2]. The main difference is that they do not include property (S3) in the definition, but make it a hypothesis in the following theorem.…”

Section: Remarkmentioning

confidence: 99%

“…Catlin and D'Angelo deduce Theorem 2.9 from a more powerful theorem (stated as Theorem 2.11 below), which is the main result in the same paper [CD2]. A key tool they use is a theorem of Catlin [C] regarding the asymptotic expansion of the Bergman kernel.…”

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confidence: 99%

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Geometry of Hermitian Algebraic Functions. Quotients of Squared Norms

Varolin¹

2008

ajm

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Hermitian algebraic functions were introduced by D. Catlin and J. D'Angelo under the name of "globalizable metrics". Catlin and D'Angelo proved that any Hermitian algebraic function without nontrivial zeros is a quotient of squared norms, thus giving an answer to a Hermitian analogue of Hilbert's 17th problem in the nondegenerate case. The result was independently proved somewhat earlier by D. Quillen in a special case, and using different methods. In this paper, we characterize all Hermitian algebraic functions that are quotients of squared norms.

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“…In the language of this section, we can phrase the following beautiful result of Catlin and D'Angelo [CD2] as follows.…”

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Geometry of Hermitian Algebraic Functions. Quotients of Squared Norms

Varolin¹

2008

ajm

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“…The proof in [6] uses the Bergman projection and some facts about compact operators, and it generalizes to provide an isometric imbedding theorem for certain holomorphic vector bundles [7]. It is worth noting that the integer k and the number of components of h can be arbitrarily large, even for polynomials p of total degree four in two variables.…”

Section: A Striktpositivstellensatzmentioning

confidence: 99%

Polynomial Optimization on Odd-Dimensional Spheres

D’Angelo

Putinar

2008

Emerging Applications of Algebraic Geometry

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Abstract. The sphere S 2d−1 naturally embeds into the complex affine spaceWe show how the complex variables in C d simplify the known Striktpositivstellensätze, when the supports are resticted to semi-algebraic subsets of odd dimensional spheres. Mathematics Subject Classification (2000). Primary 14P10; Secondary 32A70.Keywords. Positive polynomial, Hermitian square, unit sphere, plurisubharmonic function. PreliminariesLet C d denote complex Euclidean space with Euclidean norm given by |z| 2 = d j=1 |z j | 2 . The unit, odd dimensional sphereis a particularly important example of a Cauchy-Riemann (usually abbreviated CR) manifold. This note will show how one can study problems of polynomial optimization over semi-algebraic subsets of S 2d−1 by using the induced CauchyRiemann structure. Our results can be regarded as multivariate analogues of classical phenomena about positive trigonometric polynomials, known for a long time in dimension one (d = 1). They are also related to results concerning proper holomorphic mappings between balls in different dimensional complex Euclidean spaces and the geometry of holomorphic vector bundles.

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“…On a more recent date such structures have recurrently appeared in the geometric aspects of the theory of functions of several complex variables, see for instance [4].…”

Section: Final Remarksmentioning

confidence: 99%