Interface asymptotics of partial Bergman kernels on $S^1$-symmetric Kaehler manifolds

Zelditch, Steve; Zhou, Peng

doi:10.48550/arxiv.1604.06655

Cited by 12 publications

(26 citation statements)

References 2 publications

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“…, where {s i } is an orthonormal basis of H δk k . When V is a divisor, the partial density functions have been studied by Zelditch-Zhou [31,32], Ross-Singer [16] and Coman-Marinescu [7] etc. It was first shown by Shiffman-Zelditch [22] in the toric case, that there is a 'forbidden region' in which the partial density function is exponentially small.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of partial density function vanishing along smooth subvariety

Sun

2020

Preprint

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Section: Introductionmentioning

confidence: 99%

Asymptotics of partial density function vanishing along smooth subvariety

Sun

2020

Preprint

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“…This article is part of a series [HZZ15,ZZ16] studying the scaling asymptotics of spectral projection kernels along interfaces between allowed and forbidden regions. In this article, we are concerned with semi-classical Wigner distributions, W ,E N p q px, ξq :"…”

Section: Introductionmentioning

confidence: 99%

“…In [ZZ16], Zelditch-Zhou studied scaling asymptotics around Σ E of the so-called Husimi distribution of Π ,E N p q rather than the Wigner distribution. The Husimi distribution is the covariant symbol (value on the diagonal) of the conjugate B ˚ Π ,E N p q B ˚ of the eigenspace projection by the Bargmann transform to the holomorphic Bargmann-Fock space.…”

Section: Introductionmentioning

confidence: 99%

Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

Hanin¹,

Zelditch²

2019

Preprint

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Eigenspaces of the quantum isotropic Harmonic Oscillator p H :" ´ 2 2 ∆ 1 2 ||x|| 2 on R d have extremally high multiplicites and the eigenspace projections Π ,E N p q have special asymptotic properties. This article gives a detailed study of their Wigner distributions W ,E N p q px, ξq. Heuristically, if E N p q " E, W ,E N p q px, ξq is the 'quantization' of the energy surface Σ E , and should be like the delta-function δ Σ E on Σ E ; rigorously, W ,E N p q px, ξq tends in a weak* sense to δ Σ E . But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of W ,E N p q px, ξq in the interior Hpx, ξq ă E of Σ E ; interface Airy scaling asymptotics in tubes of radius 2{3 around Σ E , with px, ξq either in the interior or exterior of the energy ball; and exponential decay rate sin the exterior of the energy surface.

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“…The density function of this space, called partial Bergman kernel, appears in a natural way, in several contexts, especially in Kähler geometry and pluripotential theory (linked to the notion of extremal quasiplurisubharmonic functions with poles along Σ) see e.g. [Be1,RoS,PS,RWN1,RWN2,CM3,ZZ]. One of the motivations is the notion of slope of the hypersurface Σ in the sense of Ross-Thomas [RT06] and its relation to the existence of a constant scalar curvature Kähler metric in c 1 (L).…”

Section: Introductionmentioning

confidence: 99%

Holomorphic sections of line bundles vanishing along subvarieties

Coman,

Marinescu,

Nguyên

2019

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Let X be a compact normal complex space of dimension n, and L be a holomorphic line bundle on X. Suppose Σ = (Σ 1 , . . . , Σ ℓ ) is an ℓ-tuple of distinct irreducible proper analytic subsets of X, τ = (τ 1 , . . . , τ ℓ ) is an ℓ-tuple of positive real numbers, and consider the space H 0 0 (X, L p ) of global holomorphic sections of L p := L ⊗p that vanish to order at least τ j p along Σ j , 1 ≤ j ≤ ℓ. We find necessary and sufficient conditions which ensure that dim H 0 0 (X, L p ) ∼ p n , analogous to Ji-Shiffman's criterion for big line bundles. We give estimates of the partial Bergman kernel, investigate the convergence of the Fubini-Study currents and their potentials, and the equilibrium distribution of normalized currents of integration along zero divisors of random holomorphic sections in H 0 0 (X, L p ) as p → ∞. Regularity results for the equilibrium envelope are also included.

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Interface asymptotics of partial Bergman kernels on $S^1$-symmetric Kaehler manifolds

Cited by 12 publications

References 2 publications

Asymptotics of partial density function vanishing along smooth subvariety

Asymptotics of partial density function vanishing along smooth subvariety

Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator

Holomorphic sections of line bundles vanishing along subvarieties

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