2019
DOI: 10.1016/j.aim.2019.03.001
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Reproducing kernel functions and asymptotic expansions on Jordan-Kepler manifolds

Abstract: 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 po… Show more

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Cited by 7 publications
(11 citation statements)
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“…The Hardy space and the weighted Bergman spaces of holomorphic functions on bounded symmetric domains have been extensively studied from various points of view (see, e.g., [6,21,29]. 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.…”
Section: Jordan-kepler Varietiesmentioning
confidence: 99%
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“…The Hardy space and the weighted Bergman spaces of holomorphic functions on bounded symmetric domains have been extensively studied from various points of view (see, e.g., [6,21,29]. 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.…”
Section: Jordan-kepler Varietiesmentioning
confidence: 99%
“…Every hermitian Jordan triple V has a natural notion of rank defined via spectral theory. For fixed ℓ ≤ r let Vℓ = {z ∈ V : rank(z) = ℓ} denote the Jordan-Kepler manifold studied in [20]. It is a K C -homogeneous manifold whose closure is the Jordan-Kepler variety…”
Section: Jordan-kepler Varietiesmentioning
confidence: 99%
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“…Recently, certain algebraic varieties in symmetric domains, called Jordan-Kepler varieties, have been studied from various points of view [8,30]. Although these varieties are not homogeneous, there exist natural K-invariant measures giving rise to Hilbert spaces of holomorphic functions and associated Toeplitz operators.…”
Section: Introductionmentioning
confidence: 99%