Bergman kernels, TYZ expansions and Hankel operators on the Kepler manifold

Bommier-Hato, Hélène; Engliš, Miroslav; Youssfi, El Hassan

doi:10.1016/j.jfa.2016.04.018

Cited by 5 publications

(4 citation statements)

References 24 publications

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“…For a given Kähler potential φ, the function e −νφ(w) K ν (w, w) is closely related to the Kempf distortion function (or Rawnsley's ǫ-function) which is of importance in the study of projective embeddings and constant scalar curvature metrics (Donaldson [9]), where a prominent role is played by the asymptotic behavior as ν → +∞ (sometimes referred to as Tian-Yau-Zelditch (TYZ) expansion). For more details, see [10]. In this section, we prove our major results concerning the asymptotic expansion.…”

Section: Asymptotic Expansion Of Bergman Kernelsmentioning

confidence: 71%

“…with some η > 0, see [10] (one can take any 0 < η < 1 The asymptotic expansion as ν → +∞ (with the other parameters fixed) of the kernels from Corollary 5.3 can also be obtained by standard methods. As our main results, we will now derive the asymptotic expansion of the reproducing kernel functions associated with the Kähler potentials φ ℓ (w) from Proposition 3.6, both in the flat and the bounded setting.…”

Section: Asymptotic Expansion Of Bergman Kernelsmentioning

confidence: 99%

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Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds

Engliš

Upmeier

2019

Advances in Mathematics

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We study the complex geometry of generalized Kepler manifolds, defined in Jordan theoretic terms, introduce Hilbert spaces of holomorphic functions defined by radial measures, and find the complete asymptotic expansion of the corresponding reproducing kernels for Kähler potentials, both in the flat and bounded setting. IntroductionFor a Kähler manifold, with (integral) Kähler form ω and quantizing line bundle L, it is a fundamental problem to measure the deviation of the 'classical' curvature νω of the ν-th power L ν from the 'quantum' curvature ω ν obtained by the pull-back of the Fubini-Study form on the projective space of holomorphic sections. This relationship is usually expressed in terms of an asymptotic expansion of ω ν in inverse powers of the deformation parameter ν, known as the TYZ-expansion of the Kempf distortion function. Closely related is the asymptotic expansion of the reproducing kernel function of the Hilbert space of holomorphic sections, a fundamental tool in complex analysis. In this paper we carry out this program for an important class of algebraic varieties, the so-called Kepler varieties defined in a Jordan theoretic setting, which generalize the well-known determinantal varieties in matrix spaces. These varieties and their regular part (Kepler manifolds) are of interest from several points of view. As algebraic varieties, we show that Kepler varieties are normal, and classify invariant holomorphic differential forms of top-degree. Using a natural polar decomposition, these manifolds carry radial measures giving rise to reproducing kernel Hilbert spaces of holomorphic functions. The most interesting radial measures on Kepler manifolds come from Kähler potentials, both in the flat and bounded setting. For these, the reproducing kernel functions are related to multi-variable hypergeometric functions of type 1 F 1 and 2 F 1 . Among our main results is the asymptotic expansion of these hypergeometric functions, which in the multi-variable case is quite challenging, and leads to the TYZ-expansion mentioned above.Using chains of Peirce spaces in Jordan triples, the approach presented here can be extended to yield asymptotic expansions of TYZ-type in a quite general setting of homogeneous flag varieties. Generalized Kepler manifoldsIt is well-known that hermitian bounded symmetric domains are characterized (via their holomorphic tangent space at the origin) by the so-called hermitian Jordan triples. Geometrically, the Jordan triple product gives the holomorphic part of the Riemann curvature tensor. We use [14] as our standard reference concerning Jordan algebras and analysis on symmetric cones. For the more general Jordan triples, see [20]. Let Z be an irreducible hermitian Jordan triple of rank r. Let {u; v; w} =: D(u, v)w denote the Jordan triple product of u, v, w ∈ Z. Consider the Bergman operators B(u, v) ≡ B u,v := id − D(u, v) + Q u Q v 1991 Mathematics Subject Classification. Primary 32M15; Secondary 14M12, 17C36, 46E22, 47B35.

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Section: Asymptotic Expansion Of Bergman Kernelsmentioning

confidence: 71%

Section: Asymptotic Expansion Of Bergman Kernelsmentioning

confidence: 99%

Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds

Engliš

Upmeier

2019

Advances in Mathematics

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“…The connection between the main results and Bergman space is that a procedure is proposed to present the general form of any function in a weighted Bergman space on the Kepler manifold. Previous work covers mainly other forms of the reproducing kernel and their Tian-Yau-Zelditch expansion (TYZ expansion) [19,21]. The future work will be done in the following two parts:…”

Section: Discussionmentioning

confidence: 99%

Representing Functions in H2 on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions

Song

Sun

2022

Mathematics

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The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Because these function spaces are reproducing kernel Hilbert spaces (RKHS), three different domains on the Kepler manifold are considered and the weak pre-orthogonal adaptive Fourier decomposition (POAFD) is proposed on the domains. First, the weak maximal selection principle is shown to select the coefficient of the series. Furthermore, we prove the convergence theorem to show the accuracy of our method. This study is the extension of work by Wu et al. on POAFD in Bergman space.

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“…More recently, irreducible subvarieties of symmetric domains, given by certain determinant type equations, have been studied in [20] under the name of 'Jordan-Kepler varieties.' This terminology is used since the rank r = 2 case corresponds to the classical Kepler variety in the cotangent bundle of spheres [11] In order to describe bounded symmetric domains and their determinantal subvarieties, we will use the Jordan theoretic approach to bounded symmetric domains which is best suited for harmonic and holomorphic analysis on symmetric domains. For background and details concerning the Jordan theoretic approach, we refer to [22,25,29].…”

Section: Jordan-kepler Varietiesmentioning

confidence: 99%

Singular Hilbert modules on Jordan-Kepler varieties

Misra¹,

Upmeier²

2019

Preprint

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We study submodules of analytic Hilbert modules defined over certain algebraic varieties in bounded symmetric domains, the so-called Jordan-Kepler varieties V ℓ of arbitrary rank ℓ. For ℓ > 1 the singular set of V ℓ is not a complete intersection. Hence the usual monoidal transformations do not suffice for the resolution of the singularities. Instead, we describe a new higher rank version of the blow-up process, defined in terms of Jordan algebraic determinants, and apply this resolution to obtain the rigidity of the submodules vanishing on the singular set.

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Bergman kernels, TYZ expansions and Hankel operators on the Kepler manifold

Cited by 5 publications

References 24 publications

Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds

Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds

Representing Functions in H2 on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions

Singular Hilbert modules on Jordan-Kepler varieties

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