Convergence in Capacity

Cegrell, Urban

doi:10.4153/cmb-2011-078-6

Cited by 25 publications

(9 citation statements)

References 11 publications

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“…Then lim j →∞ ϕ[(dd c u j ) n − (dd c v j ) n ] = 0 in the weak-topology of measures for all ϕ ∈ PSH ∩L ∞ loc ( ). Theorem 3.1 is a generalization of Theorem 1.1 in [8]. As an application we obtain in Theorem 3.3 that if u j , u, v ∈ E ( ) such that u j ≥ v, ∀ j ≥ 1, lim j →∞ u j ≤ u and (dd c u j ) n → μ in the weak-topology of measures then…”

Section: Introductionmentioning

confidence: 79%

Convergence in capacity and applications

Hiệp¹

2010

MATH. SCAND.

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In this article we prove that if $u_j, v_j, w\in\mathcal{E}(\Omega)$ such that $u_j,v_j\geq w$, $\forall\ j\geq 1$, and $|u_j-v_j|\to 0$ in $C_n$-capacity, then $\lim_{j\to\infty}h(\varphi_1,\ldots,\varphi_m) [(dd^cu_j)^n-(dd^cv_j)^n]=0$ in the weak-topology of measures for all $\varphi_1,\ldots ,\varphi_m\in{\operatorname{PSH}}\cap L_{\operatorname {loc}}^\infty (\Omega)$, $h\in C(\mathsf{R}^m)$. We shall then use this result to give some applications.

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Section: Introductionmentioning

confidence: 79%

Convergence in capacity and applications

Hiệp¹

2010

MATH. SCAND.

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“…Comparing it with results from [10], we see that this is precisely the asymptotic of the limit function lim ε→0 G Iε and, therefore, in this case one has G I• = G I (2) .…”

Section: Examples and Questionsmentioning

confidence: 89%

“…This can be done easily, since the mass of G I (2) can be computed as the Hilbert-Samuel multiplicity of the ideal I (2) , and the latter equals the multiplicity of generic mappings (f 1 , f 2 ) for f 1 , f 2 ∈ I (2) , which is 12.…”

Section: Examples and Questionsmentioning

confidence: 99%

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Powers of Ideals and Convergence of Green Functions with Colliding Poles

Rashkovskii¹,

Thomas²

2012

International Mathematics Research Notices

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Abstract. Let Ω be a bounded hyperconvex domain in C n , and I ε be a family of ideals of holomorphic functions on Ω vanishing at N distinct points all tending to a ∈ Ω as ε → 0. As is known, convergence of the ideals I ε to an ideal I does not guarantee the convergence of the pluricomplex Green functions G Iε to G I ; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers I p ε converge to some ideals I (p) , we prove that the functions G Iε converge, locally uniformly away from a, to a function which is essentially the upper envelope of the scaled Green functions p −1 G I (p) , p ∈ N. As examples, we consider ideals generated by hyperplane sections of a holomorphic curve in C n+1 near a singular point. In particular, our result explains the asymptotics for 3-point models from [10].

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“…We have v j v ∈ E( ) then Corollary 3.3 in [23] implies that −ρ(dd c v j ) n is convergent weakly to −ρ(dd c v) n . Moreover, 1…”

Section: Assume That F ∈ E( ) ∩ M P S H( ) and V ∈ F( F ) Such Thatmentioning

confidence: 93%